Very interesting question though, and one I've thought about a lot, coming from Europe to the US as a kid.
It's possible that Europe may have only lost relatively. I.e. Europe may not have fallen so much as that other places have risen.
The single biggest problem with Europe may be government. There's a tradition in most northern European countries of strong central government. And the growth of technology in the 19c made it possible for a hands-on central government to be very much more hands-on. This seems to have had a disastrous effect on certain kinds of economic growth.
Remember too that Europe had two huge wars go through it during the 20th century. That can't have helped.
Right. There are tons of cool things in tech (Linux, Python, Mysql, Qt/KDE, off the top of my head) and the sciences in Europe, just that there are more in the US.
How did it lose so much so fast?