Both seem to do roughly the same thing. What, if anything, are the relevant differences?
Don't know if there are really major differences that set one far apart from the other.
One of the bigger differentiators is that Soundhound claims to recoginze songs that are hummed and/or sung by the user.
From a purely opinion oriented view, I've had better success with Soundhound at identifying songs than Shazam - but both do the job pretty well. Soundhound also seems to be quicker at returning results for me than Shazam. Again, that's just my own experience and I've, by no means, done exhaustive testing between the two.
Shazam seems to have better search but Soundhound can search by lyrics.
Here's a great video about SoundHound vs. Shazam that should help answer your question: