i feel like they are both a little ridiculous on check ins. with foursquare you can make any location a "venue" to check into. my friend has her apartment set up as well as other apartments on campus. she is a supermayor but half the places she's the mayor of aren't really "establishments"
With yelp you can't create a place but you can check in no matter how far away you are. i just tested it from about 15 miles away and it worked fine.
But the "king of d.c" has 586 dukedoms and 384 places he/she is a regular at...
that just seems outlandish.
so it looks like both make it easy to "play the game"
overall 4square is about telling friends where you are and finding you friends. and trying to obtain badges. yes it does have an explore option but its limited to your current location and only has limited info about the place
yelp focuses on reviews of these places and obtaining all pertinent information about the establishment. check ins are kind of thrown in with the app and for what it is they are well executed.
after spending a few days playing with both i find, like rosemary said, they are truly different apps. though they both provide the same service (check ins) their approach is radically different. it comes down to what you want to get out of the app. if you just want to keep tabs on your friends and also to let people know where you are. and finding new places is more of an adventure for you then foursquare is the way to go. if you have fun checking into places but that isn't necessarily the goal. and finding new places is more of an exact science where you can review tips, reviews, and information about the venue such as pricing, parking, required attire, etc then yelp is for you.