The real difference between militaries is projection power. The US can credibly project most of its power all over the world. The European military certainly exists, but it's projection capabilities are greatly less than the US', even on a per-dollar/per-Euro basis. It could defend itself against a straight-up Russian attack (which is precisely why that will not be happening anytime soon), but it leaves Europe without pieces when Russia is playing chess on the world stage [1]. If Europe would be starting to assert itself more thoroughly in the world, against credible threats, it would need to spend a great deal more money to do so.

[1]: Diplomacy is not chess, but it has chess-like elements (along with the poker-like elements). Trying to play diplomacy without the ability to threaten anything, even with no real intent or prospect of following through on the threat, leaves you with a proportionally much weaker portion.