* possible spoiler alert *
Both are puzzle kind of games. There isn't anything like the insult fighting in GF. Both storylines are fascinating and suck you right in. Great voice acting and sidekicks. I think (if I recall correctly) both games make you travel to different places.
I found that GF moved me a lot more emotionally than MI (subjective, yeah) and I think it's probably because the GF storyline is a little darker/serious (not to say it didn't have funny moments). When I finished MI, I felt accomplished. When I felt GF, I felt like I was saying goodbye to friends that I'm not sure I would ever see again. That kind of feeling.
I could go on and on, but I don't want to spoil it too much more. :)
The Curse of Monkey Island still looks amazing (if low-res), as does Full Throttle. I'd love to see the return of big-budget games with 2D art.
To me playing GF felt like reading a really, really enjoyable novel. The plot really stands up on its own. While in MI you travel back and forth different places to do various things, GF has a bit more of a journey feel to it, and you can see the main character learning and developing along the way.
Monkey Island is laugh-out-loud funny. It's very much a comedy game, with gags and a ridiculous world.
Grim Fandango is more subdued. It does have funny bits, but they're more in the absurdity of the situation. It has a lot of /style/ and attention paid to evoking a particular noir-esque mood.
it can be a bit clunky to get used to it.
despite that, while TSOMI is one of the best adventuregames of all time, grims story, characters, style and music elevate it beyond just the medium of games. this is a cultural masterpiece of that time