I have been on the Aquacat and the Sun Dancer in Belize, and toured the Belize Aggressor (which pretty much does the same itinerary as the Dancer.) Both are worth doing sometime, but I would say the Aquacat and the Belize boats and trips are not very similar at all, beyond the usual eat-dive-sleep-repeat schedule.
The Belize Dancer is physically fairly typical of many of the Caribbean Aggressor and Dancer boats (not to mention the Explorers and the Caribbean Pearl in the Bay Islands, some of which are former Aggressors.) The Sun Dancer is larger and I found the cabins a bit larger/nicer on the SD than the BA, but the dive deck larger and nicer on the BA. I found the inside eating area on the BA to be very cramped - that and the cabins make me prefer the Dancer there but nothing wrong with the Aggressor if that's not important to you. The Aquacat on the other hand is a large roomy catamaran, with way more space in general.
Belize is a good place to do a liveaboard because the best diving there is a long haul from land at the atolls - the liveaboards go out there and stay there, while if you're land based in Belize this will be a special all-day trip you maybe do one day, usually at extra expense. (If you were to stay on say Turneffe atoll you obviously dive there, and they include a trip to Lighthouse atoll, but it's still only once a week - and if you don't want to do it for some reason I think you don't dive that day.)
Most people would probably consider Belize diving out at the atolls where the liveaboards go to be better than the Bahamas, with good reefs and walls. IMO Bahamas diving done on the usual Aquacat itinerary simply doesn't have such great coral and reefs/life for the most part. It does however have lots of sharks and more varied topography and sites, with some unique dives. It also has a few more opportunities for doing above water/surface stuff, good if you're going with someone who isn't diving, or doesn't want to dive all the time. Either Belize liveaboard is usually going to do the Blue Hole, which for some is a reason to do it, for others a reason to avoid it (obviously you can sit it out if you aren't interested, but you basically lose a half day of diving.)
In January the water temperature difference between the Bahamas and Belize may be a consideration for you. Another thing for some people is that the AquaCat, being a large cat, is generally going to be a smoother more stable ride. Also the Belize boats have a long crossing to the atolls, which can be fine or can be _very_ rough, and probably more likely to be rough in the winter.
