Your assumption that there aren't many North American invasive species in Europe doesn't seem to be true, so this is begging the question. The Old World is in fact rife with invasive species from the New World. The list is long, but notable examples include:

• Raccoons, now found everywhere, especially Central Europe and Japan
• The grey squirrel is replacing the red squirrel everywhere.
• Red-eared slider turtles, kept as pets everywhere in the world and then intentionally or accidentally released.
• American mink is outcompeting the native mink species in Europe and elsewhere.
• American bullfrogs are also everywhere.
• Largemouth bass, deliberately introduced and now found everywhere.

The list goes on and on.

Of course, we can ask whether there are more invasive species imported into the Old World than the other direction. This seems to be true, and the reasons appear to be complex. In short, it seems that invasive species tend to come from areas with a long history of competition that has resulted in high genetic diversity. In other words, the invaders are more competitive; the aforementioned grey squirrel is able to outcompete the red squirrel because it's more aggressive, but it's more aggressive specifically because of competition in its area of origin.

I think this isn't really a history question, though. /r/AskScience or /r/AskBiology might be a more suitable place to ask.