Ad-Aware is an excellent program, but will not remove everything (nothing will). You should also have SpyBot Search and Destroy. You can find links to those and other helpful utilities and advice in this thread: http://www.daniweb.com/techtalkforums/thread5690.html
Please keep all replies within the thread; that way, if anyone else has a similar question, they can find the answer themselves.
No advantage to one over the other (Ad-Aware/SpyBot), you should have them both, as well as an antivirus program; keep them all updated. Ad-Aware looks primarily for adware, SpyBot for spyware, and antivirus programs for, well, viruses (though the newer versions seem to be looking for some spyware and adware as well).
Getting your Windows Updates regularly and installing SpywareBlaster and SpywareGaurd will also help prevent infections (most virsus are released after the patches have been made available).