And who is to stop YOU from building a DIFFERENT mouse trap? No one. That is competition. You just cant steal my idea for the mousetrap, but that does not give me a monopoly on ALL mousetraps.
That is the difference
No, because there's a different between copyright law and patents; copyright law is the 'you can copy the idea, you just can't copy the specific thing'. Patents give you rights over all use of the invention. So if I happened to develop a mechanism for a spring that could be used in mousetraps, and somehow managed to get a patent granted for it for such uses, if anyone else tried making a mousetrap with a spring in it I could stop them.
With a patent, you have a monopoly over the thing that has been patented. No-one else can compete with you on the patented item. They can try and compete by coming up with a product that has a similar effect but which is able to avoid the terms of the patent, but they can't compete on the specific patented item.
To give you an example from distant memory, with asprin/paracetemols(/drugs generally), they used to cost a fortune. Suddenly their price dropped to next to nothing. The reason is that for a long time they enjoyed a patent meaning only the developer could sell them. When that expired, competitors could come in, take the product and sell it on at a fraction of the price. Until that time though, the drug developer had a monopoly over the product. If they hadn't been given such a monopoly, there would have been a massively reduced potential gain to developing it, while the research costs would've remained unchanged, so the chances are it wouldn't have been developed then.