As already answered the ship anchors in Playa Del Carmen to let people off for the Mayan Ruins Tour. I've heard good and bad about this tour, but 6 hrs on a bus in Mexico didn't really appeal to me ;-( They get everyone back to the ship on this tour.
For shopping, Cozumel is the better of the two IMO. Everything is really all on the one street, whereas Grand Cayman is kinda a maze like shopping experience with higher prices. Although, the Rum Cake in Grand Cayman is incredible!!!!! I don't drink, but I tried the rum cake, and by the time I finished sampling all of it (by the 5th or 6th store, if they weren't sampling Rum Cake, I wasn't going in it ;-))), I was a little light headed when I got back on the Tender to the Ship ;-))))
Jamaica, DO A SHIP Tour!!!! Unless of course you like being harassed to buy Drugs, Sex, etc........ I did the Dunns Falls/Kayak tour and had an awesome time, completely ignored all the street people. Once you've climbed the Falls once though, you'll probably never do it again, cause it is just a HUGE tourist trap.