I don't know anyone who has exhausted everything AE or Combustion can do. Honestly, they're both great programs and can do practically anything you need - just different styles.
Overall, Combustion is quite a bit more powerful, especially since it can use After Effects plugins. But it also has quite a learning curve and a completely foreign UI.
Careful .... only a few AE plugs work in Combustion. Discreet keeps an update somewhere on the site as to which plugins work. If you ever plan on using high end compositing programs, learn combustion. It is made by the same company that makes flame systems with a similar interface, which is also very similar to shake. Bluespider is right, but you can use them both pretty interchangeably.
been using combustion since 2.0 couldn't say enuff good things about it amazing I used AE for a while great app as well its all about where you feel most comfortable from UI to different plugs etc etc.
try em both, and pick which one gels with your workflow style most. I'm an AE guy. I've tried combustion, shake, etc.. but AE fits my style and needs most (motion graphics, design, effects, some compositing, etc)