I'm wondering how they really compare. Here are the facts I've managed to find so far:
The Economy of the EU (the largest economic block in europe) is 25% larger than the USAs, but the EU's population is about 40% larger.
The GDP per capita of the developed countries in Europe (including those outside the EU) is roughly equal (about 45k dollars) to the USA. Europe as a whole (i.e. with the former communist countries) has a significantly lower GDP per capita.
The average developed European wage is higher than the average wage in the USA due to higher income inequality in the USA. However, the purchasing power of americans in terms of housing and consumer goods (including food and fuel) is much higher.
So overall an average american worker can buy more stuff and have a bigger house.
GDP per capita is higher for the most advanced states in the USA than any of the large economies in europe. But the wealthiest smaller states in europe (i.e. norway, denmark) have GDP per capita significantly higher than any US state.
Again in terms of buying stuff (purchasing power for consumer goods) the richest states in the USA come off better than the richest nations in europe. However, those nations have the most comprehensive social benefits in the world.
In terms of exports. The EU is the worlds largest exporter of manufactured goods. Germany on it's own exports more than either the USA or China.
However, in terms of high tech I would give the USA the lead.
In civil aviation europe and the us are roughly equal (boeing vs airbus). But the USA has a huge military aviation sector.
In pharmacuticals the two areas are about equal.
In electronics the USA is clearly in the lead. There are far more european designed microprocessors in the world (ARM) but all the higher power designs are american (intel and amd). And the USA is the home of the worlds important software companies.
Europe has more of the worlds largest corporations. But not in terms of population, again the USA is in the lead here.