Fanboy vomit post incoming! I haven't been able to write about MI for a while...
While I love Monkey Island 1 and its pure simplicity, I definitely fall in the "majorly prefer Monkey 2" camp.
I think the writing is stronger, the storyline feels more deep and original than the first even if in many ways it's structurally identical to Monkey 1 and every other well-made 3rd person point and click from that era. I really really like that the story delves deep into the lore of its own universe, with the history of the four map pieces and the legend of Big Whoop, but it does it without turning the game into a continuity bog-down "uncover the past" festival - Monkey 2's storytelling just has a great strong undercurrent to it... if that makes sense? It feels like there's a ton going on just beneath the surface, that the world you're playing in is very substantial and there are events going on stretching well before you got there, but it's done without getting in the way of Guybrush's piratey adventure and without stepping on the jokes. I think the game is just very well balanced on almost all sides from an overall plotting/storytelling/dialogue perspective. It feels adventurous, it feels big and epic, it feels old, and it somehow manages to be genuinely hilarious and occasionally very dramatic, sometimes seemingly at the same time.
The artwork in 2 is amazing, and though the game's 320x200 256 color technology hasn't held up, the background paintings themselves and the art direction by Steve Purcell and Peter Chan are still some of the most engrossing and atmospheric pieces of game art I've seen (and their general visual aesthetic and color palette were amazingly basically lifted wholesale for the first Pirates of the Caribben film - possibly unintentionally, but its nice that MI 2 was 15 years ahead of the curve on that look). Though next to nothing was animated in those games due to technology and budget, the world of Monkey Island 2 still seems amazingly alive and real to me - moreso than most games I've ever played.
Granted, it does have the infamous monkey wrench puzzle among others, but IMO that's par for the course for games of that era, and therefore MI2's handful of hideous logic leaps are forgiveable. The game doesn't allow you to accidentally walk off cliffs by clicking on them, for instance, nor does it have you sticking scotch tape on a hole to get some cat hair which you combine with syrup to make a mustache to immitate a man who does not in fact have a mustache. Worse things have happened in the history of adventure gaming than MI2's one or two overly-obtuse puzzles.
Monkey Island 2's ending, from a purely "make me feel good about myself at the end of the story" fulfillment point of view, definitely stinks, but I think it totally holds up thematically. I think the first two games make plenty of soft allusions to where the end of the game is headed, and though when I was 12 it confused me more than anything, at this point I have a huge amount of respect for the end of MI2 and enjoy it quite a lot. I think it's the closest you can get to pulling off a great David Lynch style thematic but not plot-based wrap-up in what is essentially a comedy/adventure cartoon. Hats off to them for having the balls to do it, and double hats off for successfully pulling it off (as long as the player is willing to check their expectations at the door).
As for Guybrush and Elaine, and Monkey Island 3, I think where they ended up as characters throughout Monkey 2 is really good and far more accurate than where the writers of Monkey 3 made them end up. The way they end up in 3, to me, seems overly trite and predictable, and I think that the ending (and, come to think of it, overall plot) of Monkey 3 was only allowed to exist because the writers of 3 grossly dumbed down both Elaine and Guybrush.
In 1 and 2, Guybrush is a bit of a goof, yes, but it's his spirit of adventure and desire to never give up - rather than, as the writers of Monkey 3 would have you believe - his desire to be a lackluster do-gooder, that drives him through the story and makes him an interetsing character. The same with Elaine - in Monkey 1 and 2 the character she most closely resembles is Marian from Raiders of the Lost Ark - a bit immature at times, maybe a bit bitchy/spoiled, but by and large her main strengths as a character lie in her being an independent woman with a quick wit, able to not just play with the big boys but often beat them at their own game. In Monkey 3 she was reduced largely to a whiney demanding woman with a poor posh British accent who complains about jewels and punches Guybrush in the face. This gross misinterpretation of the two main characters got even worse in the fourth game. I say "boo" to that.*
I don't want to spoil any big plot bits for Melanie, so I won't go any further than that, though my internal nerdometer is telling me I've gone too far already.
* I guess it should be said that despite my extreme fanboy nitpicking and dissection, the things I brought up don't actually diminish my enjoyment of Monkey Island 3. It's a great game. I think it rightly earned its place as one of the true classics of the adventure game genre. I just also happen to think that their portrayal of some previous characters and themes was a bit weak. 
