There is so much in MoM it's gonna be hard to cover it all but I'll try to be brief as well as descriptive so you know what I'm referring to.
Leader Customization
This is one of the places MoM really shined because based on what you picked would determine a lot of how you played the game. You could choose between 5 Schools of magic Nature, Chaos, Life, Death, and Sorcery. Unlike Elemental you could take up to 11 Books in each area and the number of books you took determined how many spells you could take from each line. The spells you got were randomly chosen so you were never sure until you researched up the tree which spells had been unlocked for you unless you got enough spell books to ensure you got them all.
Besides spell books there was several traits which also could drastically effect how you would play the game. For example making magic items cheaper, converting gold to mana cheaper, more merc offerings, cheaper mercs, more mana from nodes, more experianced units, and etc. Those are the ones I can recall off the top of my head but they could really have an impact on the way you play the game.
Race/Faction Choice
Unlike Elemental which only had the 2 factions there was about 8 races in MoM as I recall. Picking you race only impacted your starting city as the population was all of that race. Each race had different advantages when it came to farming, production, gold, and mana production. While they did share a lot of the basic buildings they also tended to have buildings that only a few races could unlock. Combine this with the fact that each race had it's own unique units and it really added a lot of flavor to the game.
Part of the reason for the neutral cities littering the map was so that you could capture them and building other race's units if you wanted a more mixed army. Since your racial pick only determined the starting city and any settlers built in a city are the same race as that city. Or if you wanted a more purist setup you could simply raze the neutral cities and build your own in their place. So racial pick could be important or not depending on your play style.
Magic
While you got to pick from 5 book areas at the start there are actually 6 branches of magic. The 6th is Arcane which all players have access to all spells in. The number of spells in that area though are fairly limited and consist of a few universal spells such as making magic items and creating units to capture nodes.
Each of the 5 main branches you could get books in consisted of 4 levels (Common, Uncommon, Rare, Very Rare). Each of these levels had 10 spells in them, so when included with the spells from Arcane total over 200 spells in the game. And each of the spells tended to be fairly unqiue. Even though spells could do the same thing a lot of times they did it in different ways. For example one spell may do X amount damage to a Unit while another spell does Y amount per model in the unit. So if you think of squads in Elemental instead of the spell hitting for 5 damage it would hit all 4 guys for 5 damage (aka 20) while an individual unit would only suffer the 5. This is just one example of how they help make each spell unique.
You could cast global spells for upkeep which could drastically effect the game. Such as prevent travel from one world to the other, all enemy spells have a chance of fizzle, spells from curtain areas cost more to cast, summon units get boost, and so on. These spells were often expensive to cast and had a high upkeep but they could drastically change the flow of the game.
Summon creatures from each of the areas were fairly unique with their own abilities. They cost mana to upkeep and were often slightly better then normally training units. They were however unable to level up so they were stuck at their strength and a lot of times fully leveled up units could out preform them.
Enchantment spells could often be cast in and out of combat. Out of combat it was permanet with an upkeep cost. In combat it was a fraction of the cost but only counted for that one battle and went against your limit of caster for that battle.
Global Tactical spells which were kinda like minor versions of the global spells except they worked for just one battle. These could also be very important and turn the tide of battles.
Magic use though was pretty simple and straight forward though. You earn mana through income similar to gold in most 4x games. Which you then use to cast spells on. How quickly you can cast spells was determined by you skill which was a stat you could slowly increase by putting some of your mana income into. The mana income was split over 3 areas. Skill, Spell Research, and Mana Pool (For casting). The higher your skill the more of your mana pool you could draw from each turn.
So say it cost 300 mana to cast something. Then someone with 25 skill would take 12 turns to do it while someone with 50 skill would only take 6 turns. Also when it came to tactical combat that skill determined how much you could cast in battle which also drew from your global mana pool. So if you had a high skill but no mana you couldn't cast anything where as if you had a low skill but tons of mana you could barely cast anything as well. So it became a balancing act as too much skill and you'd quickly burn through your mana reserves if you were not careful.
Also if you didn't focus on research then you you may have a ton of skill and mana but only be able to cast weak low level spells.
Item Creation
Creating an item in the game often required a lot of mana but you got to pick what kinds of bonuses you wanted on it. +ATK, +DEF, +RESIST, +MOV, Magic Effects, and so on. The magic effects you could place on an item depended on what spells you knew. So you could not make a ring of regeneration unless you knew the regeneration spells. Also the "type" of item determined what bonuses you could put on it. For example only staves and wands could hold spells which when equipped to a caster hero allowed them to cast that spell for free each combat X number of times where X was 1-4 based on how much you choose when making the item. The more you added to an item the more expensive it got on an exponential scale. So like +1 Atk cost 100 Mana while +2 cost 250 and +3 cost 625. So some really good items got expensive fast.
Heroes
Heroes had 3 equipment slots which determined what type of stuff they could equip. Depending on the class of the Hero they could have range(bow), melee, and/or magic. Each hero often had unique traits which increased as they leveled up. Properly equipped heroes could be fairly nasty but sadly just like in Elemental a lot of people found them a bit lacking because the cost to equip those magic items was so high it was often cheaper to summon several high end magic beast then it was to make them the expensive equipment they needed. Of course the plus side was when a hero died you got to keep all their magic items so it wasn't a total loss. And often times it's more the items that make the hero.
Units/Combat
As I mentioned in Race section each race had it's own unique units. Though there was some similarities such as every race had spear men and swordsmen they would be completely different. The number of models in each unit and the stats would vary. For example trolls which were big had 4 models in a unit while the halflings had 8. And the stats on them different as well so like the trolls had more HP per model thus it would take more damage before thier unit would diminish but at the same time when it did it would take a bigger % of it's offensive power out. Also like I mentioned before some spells work better against takes with lots of models in the unit.
There was a lot more to combat then simple swing and hit. Obviously placement on the map played a huge part but also the various abilities of creatures could play a huge role. For example combat between 2 units happened at the same time so it was possible that both units would die, though this was rare. So abilities like first strike were pretty valuable, in elemental's combat all units take first strike. Invisibility not only gave a defense bonus but also made it so range units couldn't target you. Flying units that only range units could hit. Weapon immunity meant that only magic weapons or magic spells could harm them. Regeneration so you heal during combat. Ammo and Casting was limited so no making an army of archers and parking them at the back of the field picking off the enemy, most guys got 4-8 shoots per combat.
Units leveled up with a total of 7 levels. The first 5 anyone could reach but in order to get the last 2 you needed a global spell which increased your unit's max level by 1 and the Leader trait which also increased your max level by 1. This was a pretty nasty combo as it made normal units quite a bit better then summon units. Each level increase gave you some bonus to either ATK, DEF, RES, or HP. So there was a noticable difference between a fully leveled up unit and a new one.
Without going to deep into the mechanics of it combat in MoM was handled with six sided dice. How ever many ATK you had was how many dice it got to roll. You needed a 4 or better to get a "hit" and then the enemy would roll their DEF value in dice which would show how many hits they negate. Anyone who's played table top war games should be fairly use to this concept. There were ability like Lucky which the halflings got and it made it so they succeed on a 3 instead of 4, so instead of 50% chance they had 66% chance. Their were also abilities which decreased the success rate. But in general if you had a guy with 4 attack that ment you were rolling 4D6. This was pretty good at giving more consistant results since all hits or misses were fairly uncommon.
City Building
Each building in the game cost upkeep so you had to choose what to build carefully. After the basic increase of production type buildings the rest of the buildings were mainly for unlocking units to train. Since each race had their own units they also often had their own unique building tree. Like some races could build the fighters guild while others could not. But the units that came from it were still unique to that race. So there wasn't really a lot of different building types but more like what they got from them was more unique based on the race. Even the production buildings though not every race could build. Like some of the high end mana producing buildings not all races could make.
But because of their unique build trees and unique units each race really felt unique and so did their cities. This really add flavor as well as strategic options to the game since sometimes your play style best fits with a curtain race types building options.
As for city management though you manage your population the same way you did in Master of Orion 2 by dividing them up into farmers/industry. The land around the city gives bonuses to them so location is still important. And like other 4x games you build settlers which go out and found a new city though the race is determined by the city the settlers were built in. Also when you found a city it starts out as an "outpost" which doesn't produce anything and you have no build options. You must wait until it grows enough for it to actually be a city and then you can build at it.
Diplomacy
Frankly this area of the game sucked. It was pretty standard diplomacy for the time. But given how much diplomacy options have improved over the years in 4x games it's understandable. Nothing really new or inovate here that I recall. It was standard for it's time but figured I'd mention it for completeness.
Map
It was the first to have two worlds. A lot of Fantasy 4x games have copied this idea since then. Some using Light/Dark while others use Above ground/Underground as the reasoning for the two maps. As stardock pointed out though this was originally done due to hardware limitations as a way to increase the map size. Because when you compare MoM to most modern 4x games its map is very small.
Mana Nodes
The map had mana nodes littered around which when captured would give you mana each turn. They came in different types depending on the different schools of magic. Curtain Leader traits increased the amount of mana you got from curtain types of nodes. This ment if you had choosen though traits then capturing the nodes because even more important to you.
I'm not sure how the new magic system is going to work in Elemental. But with the global mana pool it sounds like shards might be moving more towards the way the nodes worked in MoM.
Closing
I think I got most everything. While I know I could of gone into more detail in some areas, such as the magic system, I think the general concepts are there enough to get an idea of what MoM was like. But as you can hopefully see from this break down there was a TON of different options you could take when approaching how you wanted to play the game.
As for some of the people that complain about the AI. Yes in release day v1.0 the AI was worthless. But after the patches and other work the AI in 1.31 is somewhat decent. The biggest problem in terms of balance stems from some over powered combos that players can easily exploit but the AI rarely does. If you take 10+ books in one field you get to choose a very rare spell at the start of the game. Most people choose the best summons from that spell line. So basically they start the game being able to summon the best creature in the game for their spell line. This easily can stomp any of the early troops it might encounter. Even if they don't pick the best in their line a lot often pick summons with weapon immunity something early game stuff can't counter easily.
If you avoid using the cheap rush tactics against the AI then often times you will be in for a fairly good match, expeically at the harder difficulties. MoM doesn't have any R&D outside of spells and relies on the fact that you need to build the pre-req buildings to get high end normal units to slow the pace at which high end stuff hits the field. The starting spell picks though allow you to get around that and I think could easily have been redesigned so you just get most or all of the lower level spells instead of giving you none of lower levels ones and a pick of very rare when you take enough spell books. That one simple change would negate like 90% of the cheap unbalanced rush tactics.
Besides I often found it more useful to have a bunch of the lower level spells then 1 high end at the start, more versitility. Your pretty much locked into the use all mana at the start to slowly summon the big summons, send it out to start claiming neutral cities, while summon another and so on. Then when you find first 1-2 wizards quickly stomp them. But that point you have such a massive empire even though the other wizards developed enough to put up a fight against the lone high summons monster you have so many cities and units in general it's overwhelming to take them out. Though sometimes if there is one of the other world that's manage to capture all of it that can be a fairly interesting fight.
EDIT: For a quick Civ comparision on the rush tactics. It's like if at the beginning of the game you can build tanks while they are stuck with warriors and archers. It maybe take you 20-30 turns to build that 1 tank and them 3 to build a warrior but in the end their is little chance of an army of warriors winning against that tank. And they can't get their units to you before you build it so their is not real counter rush other then them building their own tank. Plus combat is like E:WOM where who armies fight and not like Civ were it's 1 vs 1. So the tank can while out that whole army in a single turn.