So how, in your opinion, does it compare to the first one?
It's more like an expansion pack than a sequel. There are a couple new features (double height screen --- a lot of people found the gap between screens annoying, but I actually kinda like it, I feel like a ninja when I shoot something in there --- and you can switch between different babies for Yoshi to carry on his back, which has been used wonderfully in a handful of levels but most of the time doesn't add much) and a couple missing ones (no items/inventory, no watermelons, minigames with special control are replaced with challenges more closely tied to the main gameplay --- an improvement really). The music is all different, and worse, though I don't think it's as bad as it's made out to be --- it just really suffers from comparison to the original.
Other than that, it's basically the same game with different levels. Same setup with the timer and egg system, same controls and moves except for the new baby abilities, same level select interface, same collecting red coins and flowers, mostly same graphics, mostly same enemies and interactive level elements. The level design itself, like I said, runs the gamut from thuddingly mediocre (most of the first world is in this category, so if you play it, do at least get through some of world 2 before writing it off) to superb. I'd say the highs are on par with YI SNES's, but the lows are lower. However, the last world and the bonus levels are supposedly shit-hot (as well as shit-hard) and I haven't done those yet, so I can't give my FINAL JUDGEMENT.
Oh, and they added cutesy vocalizations to Yoshi's actions a la Mario 64. Which was really off-putting at first, especially with the (also cutesier) music, but I got used to it.