The night life on Cozumel is there, though somewhat tame compared to Cancun (which is a good thing, IMO).
Carlos n' Charlie's and Señor Frogs have gone upscale, are in the Puta Langosta complex, and are geared to the well-heeled crusero. You'll pay something like 7 bucks for a margarita and 4 bucks for a beer, and the atmosphere is as authentic as any mall bar in the US.
Joe's Raggae Bar was at one time my favorite night spot on Cozumel, but it closed down about 4 years ago; it has since reopened on the waterfront upstairs next to Palmeras (on the north side). It used to be a little hole in the wall back away from the main drag with cheap beer, good local bands, and not many turistas. Now it's on the main drag in what has to be an expensive place for them to lease, but in all fairness, I haven't been there yet.
There is Neptuno, an authentic 70's style disco, complete with red velvet and brass, across the street from the Acuario restaurant at the traffic light on the south side main drag.
There is Fat Tuesday's, on the north side of the plaza about half a block back from the water. It's an outdoor bar with loud gringo music, limbo contests, and drunk turistas gringas dancing on the bar.
There is Hard Rock Cafe; if you've ever been to one, you'll find this one identical to whichever one you've seen. There were both a Hooters and a Planet Hollywood on Coz at one time, but they folded - good riddance.
There are lots of small bars around town where you can get a beer at a reasonable price and socialize; when I go out at night on Cozumel, that's what I do. Your hotel (wherever you are staying) may have a bar as well.
Oh, BTW, you cannot buy alcohol in stores on Sunday (though the bars are open), so if you are stocking a cooler in your room, plan ahead for that. It may or may not be an all day ban; I cannot remember. The liquor stores are closed all day, anyway.
