When a movie or cartoon property gets licensed today, you generally get one developer doing a fairly uniform port of it across various platforms. Not so much back in the '80s and '90s, where often each port of a game would be given to a completely different publisher and licensed games could vary wildly between platforms.
With Aladdin, Disney wanted games for the SNES and Genesis. They licensed the Genesis version to Virgin and the SNES one to Capcom. They appear to have laid down some general guidelines -- it has to be a platformer, Ali hucks apples, there's a genie-related bonus game in between levels, and it roughly follows the movie plot -- but otherwise the two companies were free to develop their own unique games.
While both are platformers that recount the events of the movie with a few extra elements thrown in for padding, Capcom's approach is a bit slower-paced and has a different, more precise and acrobatic feel to it. It plays somewhat like some of Capcom's licensed arcade platformers (such as Little Nemo and Willow) mixed up with the mechanics of Prince of Persia. Enemy avoidance is prioritized more, as Aladdin does not have a sword in this one and his thrown apples only stun most enemies for a couple of seconds (though they do outright kill the tinier enemies). You can also jump on most enemies heads to knock them off the screen in this game, where that would just get you damaged in the Genesis version.
That's not to say Capcom's version is significantly easier than the surprisingly challenging Genesis game. It starts out easier in the first couple of levels, but the difficulty gets jaggy from there. The main sticking points are a couple of Battletoads-like auto-scrolling levels where you're riding on the magic carpet and collision with anything kills you instantly. But some of the later levels also have series of pixel-perfect jumps sprinkled here and there that can be a real buzzkill if you're not scumming save states. On the whole, I'd say the two games pose about an equal challenge on Normal difficulty, and will take some dedication and level memorization if you aren't playing with save states. However, it's also easy to level-skip in this one with a simple password system.
The only major problem I had with the Genesis version is that the level design felt cheap at times, with enemies sprung on you very suddenly after scrolling the screen, or outright hidden behind background objects. Outside of the irritating magic carpet level, I think the SNES version actually does the platforming aspect a little bit better. The downside is that it doesn't have the same gorgeous attention to detail in the art that the Genesis game did (one of the rare cases of a Genesis port looking markedly better), but that's not to say it's an ugly game. Ali is animated pretty well and there are some neat backgrounds en route. It's just not as consistently awesome from end to end as the Genesis art, but overall the platforming is a little more solid. No huge surprise there, as Capcom had long experience in this genre while Virgin was a relative newcomer to it.