The application process aside, it really depends on whether you want to sit in a control tower and look at aeroplanes while you control them, or sit in an area control centre and never see the aeroplanes you're talking to.
Eurocontrol is solely area work, whereas at least with NATS, there is the possibility of working at an airfield (although only a few from each course get this option - the majority do go on to do area control work).
Eurocontrol are a very good employer, and the benefits provided are also very good.
Looking at some posts you may think that NATS are an awful company to work for - well, they aren't perfect, but they're a damn sight better than some I could name!