There are no differences if we are talking about the game - same items, etc. EU just gets patch with one day delay (with exception for Trundle patch, where EU got it first!!)
There are "community" differences:
* NorthAmerican (not just USA -.- What about Canada? It's not small one) server has more expanded forums and more staff members.
* Game play style ("metagame") may be a litte different. Looks like NA focuses more on freeroam, 2-3 people fights. EU mostly focuses on mid or jungle wars with 5 people.
* Champion "styles" - I have heard of NA and EU Ashe style - item difference. One rushes to IE, other GhostBlade.