Staying in downtown Cozumel means that you'll be located in close proximity to pretty much everything the island has to offer. You'll be within walking distance to two supermarkets (Mega and Chedraui), plenty of restaurants and bars, small shops, the ferry to Playa del Carmen, and the beautiful town plaza.
This area of the island is located just five minutes from the airport, so it's super convenient if you're flying into Cozumel and have an early flight on the way home.
The "In Town" or "Centro" neighborhood can be roughly defined by the boundaries of airport road on the north end of town (the road you'll most likely take when you arrive at Cozumel airport), 11th Avenue (the division between Centro and Corpus Christi) on the south, and 30th Avenue, a main commercial avenue, to the east.
If it's your first time visiting the island and you'd like to experience a bit more of the local, as well as touristy, side of Cozumel, staying in town is a good option. During the day, the area gets busy with cruise ships passengers walking around and taking in the sites. This is especially true during the high season months (roughly Nov to Apr) when the island receives the most cruise ships, somethings up to 10 per day. Luckily, the ships leave by the end of the day so the downtown area empties out by around 5 pm in the afternoon.
It's at this time, and especially closer to sunset, that you'll see many locals comes out to enjoy the sunset and the activities around the central square during the evening hours. On the weekends, the city organizes dances in the central plaza, often with a live band. And during the weekdays, the large modern fountain that's embedded into the concrete of the central plaza, comes alive with jets of colorful water shooting up into the air. It's a great time to wander around, take in the sights, and bring the kids to run around and burn off some energy before bedtime.
The shorline in this part of the island is largely rocky and not great for subathing or swimming. While there are some areas with sand, there's a fair amount of boat traffic in this area, especially during the day, so it's not a great spot to catch rays. You can certainly get into the water - it's very clean as it is in all parts of Cozumel - and some people do, but largely a bit south and north of the main part of downtown. Both to the north and south you'll find a few beach bars that are great spots to take a quick dip.
From this part of the island, you'll also find the east (or "wild" side) of the island really accessible. The main road that goes to the other side (it's referred to as the wild side because there is no electricity on that side and very little development in general) cuts right through the central plaza. In about 20-25 minutes by car, you reach the east side and enjoy the beautiful, largely uninhabited beaches that stretch the whole lenght of that side.