They are very different islands.
Isla Mujeres is a lot smaller, closer to Cancun and is more a party/beach island, which attracts many young backpackers, snowbirds and day-trippers. Whereas Cozumel is a whole lot quieter (except for the Malecon and Plaza areas), and attracts mainly divers and cruise-ship people.
Both have their charm. I like Isla for the dancing, live music, nice beaches, partying…
I love Cozumel for the diving and the general "healthier" way of life.
I only dove once near Isla Mujeres (Manchones) and was disappointed : lots of fish but barely any reef. Very shallow and no current where I went, but I don't know if that's the case at every site.
You can always do an overnight trip there from Cozumel to see if you like it. Better yet, if you fly in / out of Cancun, Isla Mujeres is a great place to spend the first or last night of your trip. It's a 15 minute boat ride from the continent. Otherwise, it's less than a 3-hour trip each way if you make all the connections (Coz ferry + bus Playa-Cancun + taxi Cancun-Puerto Juarez + ferry to Isla).
There are many lodging options on the island, either in town (the North tip) which allows you to walk everywhere (beach, restaurants, shops, ferry…

, or further South, which is a lot quieter (but you might then want to rent a golf cart or rely on taxis, as there is only one bus route and it is not very convenient).
What else ? The beaches are awesome (although Playa Norte, which used to be an absolute wonder, has greatly suffered from Wilma as well as the building of big hotels), with extra fine white sand and cristal clear turquoise water. The snorkeling is nice as well and they offer trips to see the whale sharks if in season.