Shazam works much better than Midomi in my incredibly informal, anecdotal experiences... never got any of the extra midomi features to work when I could use them, either.
I think it's pretty well established that Midomi is faster and has better music recognition than Shazam. There's a few video comparisons out there that show this:
Here's a brief description about its proprietary search technology.
http://sev.prnewswire.com/multimedia-online-internet/2007012...
Here's an article that compare the features of both iPhone apps.
http://www.theiphoneblog.com/2008/11/05/app-app-shazam-midom...