Are you already staying somewhere? That sort of drives (pun intended) the decision because it's about an hour drive between the 2 areas.
Arguably the better diving is NorthEast. The west side operators don't often go out there except some do an extended 3-tank. Operators like Ocean Frontiers on the East End dive the entire East side. Babylon and several sites near it are the ones to dive if possible. Or sites off the North Wall typically accessed from the west side. It's all about the distance the boats have to travel from their home dock - and maintain their daily schedules.
Another factor to consider is weather. There's the highest likelihood that the North Wall will be undivable during the winter months. If that happens the westside/NW operators move boats south and shuttle you down. If that happens to Ocean Frontiers, they just stay south instead. Unless there's a storm over Cayman, generally the better west side dives like Big Tunnels, Orange Canyon etc. are generally divable even if the north wall isn't. That's a generalization of course.
The other consideration is that on the East Side there's 3 dive options, Tortuga Divers(Red Sail) at Morritt's/Rum Point, Ocean Frontiers/Compass Point. Of course all are walk-ins also. I've heard Tortuga typically sees a less advanced crowd but IDK for certain. So you may want to check availability also - my buddy couldn't get a space with Ocean Frontiers once a couple years ago.
If your plan is to dive Stingray City or the Kittiwake, those are more easily accessed from the West. Except Red Sail also does it from Rum Point so you could drive there from Morritt's etc. I believe Ocean Frontiers asks you to drive around and meet a boat - either in Georgetown or at the Sound marina for those trips.
Another consideration is that the majority of dive operators are either based around Georgetown or farther north in one of the Seven Mile Beach marinas. So if you like someone besides the three I listed above, they're all west side based. Or in Georgetown - not my favorite idea when all the cruise ships are in port daily. Besides the congestion it can possibly affect dive site selection due to schedules/experience levels.
For snorkeling, some of the sites
I know of are on the West Side - Smith's Cove, Cemetery Beach, Turtle Reef, Lighthouse Pt. etc. The last two may be influenced by the north side weather also although often one or the other is still accessible. Testudo does a nice job with his guide for that:
TESTUDO'S SNORKEL GUIDE Cayman Off the Beaten Path