Now this is certainly a gross simplification, but here's how each might apply to computers:
Electrical Engineers - How do we create faster gates? (MOS, BJT...etc)
Computer System Engineers - How do we create faster cpu's using those gates?
Computer Science - How do we create faster and more useful programs to run on those cpus?
Each one builds on the others, and there's quite a lot of overlap between them. I was in a simliar situation to you and I chose the EE route just because it was a better program then the CSE. And of course, even if you get your degree in one thing, you might wind up doing something completely different.