The Mayan is about 25 years old, while the Sands is about 8 years old. HOWEVER, the Royals are constantly maintained and upgraded as necessary, and the age doesn't really matter.
The Sands does have some enhancements over the Tris - e.g. I think the kitchen countertops are solid surface or granite instead of laminate, and more upscale appliances, and a big indoor activities center (compared to a smaller one at the Tris). On the other hand, you'll have more pool areas and more restaurants at the Tris. You can't go wrong either way, and the free shuttle between the Sands and the Tris runs every half hour so either way you can spend an afternoon or a day at the other.
I'm a fan of the Tri-Royals - hence I'd go for the Mayan. I'd also strongly recommend the two bedroom, so the kids and their stuff aren't underfoot in the living room, and so you don't all have to share the same bathroom and closet.
Do you mean cab fare from the airport? Unless things have changed, you can't actually take a cab from the airport - instead there are a bunch of white vans that seem to have the monopoly on transportation from the airport. It was around $10-$11 per person last time I looked. A cab back from the Royals to the airport is less b/c the cab doesn't charge per person.
I strongly recommend pre-booking your transportation from the airport, either via Thomas More (royalresorts.com, down on the left side) or Brant Boston (entertainment-plus.com). Both companies are highly recommended. The services are practically identical - same type of van, same or at least similar price. An advantage of Thomas More is that your transportation will be billed to your resort bill, so you don't have to worry about dealing with money in the airport. If you don't prebook, as you leave the airport you will see a kiosk that sells transportation. I'm pretty sure the price will still be about the same, but it's a lot more convenient to have it already arranged.
VERY IMPORTANT - as you walk through the airport, you will be accosted by many people asking you where you are going, are you going to the Royals, do you want Thomas More, do you need a ride, etc. Do not stop, do not talk to them, do not let them help with your luggage, do not sign anything, do not give them money, do not think "oh, good, here's my Royal Resort transportation." They are the timeshare sharks, NOT your transportation representatives.
Oh yeah - I'm with Geoff on the rental car - not necessary, I wouldn't want one even if it was free, because of crazy drivers, traffic, parking.... However, opinions vary, so you might want to search this forum for car rental to read other perspectives.