Cozumel has the advantage of being an island off the mainland. It is more tourists than anything else so they don't tend to have the problems you see on the mainland. We have been to Cozumel a dozen times and never felt uneasy. I'm not saying they are crime free but it no worse than going to any decent sized city in the States. If you practice personal safety like you would if you go to a major city you will be fine.
As for what to do there, it is known for its clear water, scuba, snorkeling or just laying on the beach is great. Or you can see Mayan ruins, do a dolphin experience tour.. you name it and they have it. It really depends on your interests.
For Belize, they are big on their cave tubing. They have a river cave system and they take you on guided tube floats through the caves. They also have Mayan ruins. They have a "jungle tour" where you can see onkeys and such. They also have zip lining. Belize also is big on scuba and snorkling since they are on the worlds 2nd largest barrier reef system. But you will have to take a excursion to do that since you need a boat to get to the snorkeling sites.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013