Hi DSR, we have been to both ports in the past and will do again this Feb. In Grand Caymen we have done the Sting Ray city snorkel twice and also enjoyed 7 Mile Beach as well. You can take a taxi to many locations that offer lounge chairs for a small fee and usually can find some food and drink as well. also, shopping is close to where the tenders dock. In Cozumel we have made it a shopping day (wife likes gold and bling) and do the strip for shopping and haning out. Penny will likely chime in with some of her great beach favorites. Enjoy !
In Cozumel, I went to Paradise Beach the last time I was there. There is no entry fee, as the impressions are you'll spend money while you're there. There is a $2.00 fee for lounge chairs, and for $12.00, you have unlimited use of their aqua park, floating beach mats, and snorkel equipment. There is also a restaurant and bar on the premises.
I've heard good things about 7 mile beach in Grand Cayman, but haven't been there yet. Maybe one of my future cruises if I go to Grand Cayman I'll check that out. There is shopping in Grand Cayman right at the tender area, but that area I have heard is the most expensive. For bargains, you're better off walking further into town.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)
Hi DSR, Cozumel has some great beaches. Chankanaab is my favorite because of the snorkeling. Paradise is very similar to Chankanaab, but doesn't have good snorkeling or a zoo or as many shops, but does have a nice beach with chairs, restaurant/bar, and water toys. Nachi Cocom is very nice, but you must have reservations, they only allow 100 guests. They have a swim up bar, restaurant, and a nice beach area. (We stopped there to get out of the rain, so didn't check out the snorkeling). There's a Carlos 'n Charlies, and many more on the way down to the southern end. Chankanaab and Paradise will be the least expensive to get to via cab, Nachi too. There's plenty of shopping at either port there, too.
Grand Cayman my favorite beach is 7 mile. There have been some good posts for Rum Point, but we haven't been there, yet. The snorkeling is really good in Grand Cayman. There's a dive shop (Abanks) that used to let you go into the water there, plenty of wrecks right by the tender area as well as coral and sea life. Stingray City is the main attraction there, which we've done a couple of times. I'm just as happy taking a bus done to 7 mile and spending time there or snorkeling. It's a shopper's haven there, too.