Traveling with a 3 and 4 year old children. Would like to rent a house.

There's a lot of houses for rent on Caye Caulker and it's laid back attitude may be just right for you. They have a web site that shows all of the rental units. Just do a search for Caye Caulker Belize rentals.

Placencia probably has rentals too and is on the mainland so you don't have to ferry or fly to see the ruins, jungle. You can rent a 4 wheel vehicle at the airport and have the freedom to go where you want when you want. It will also allow you to saty anywhere on the peninsula and still go to town for meals/shopping. It is a long way from Placencia to the heart of the ruins though. You can get to some of the smaller sites with a car but to get to the larger ones, you either have to take your rental car for a full day trip or go with a tour company. The Jaguar sanctuary is fairly close tough.
I think if it were me, I'd either break up the vacation and spend a few days somewhere around San Ignacio for the jungle/ruins activities and then go to the Cayes or I'd go to Caye Caulker and take a few day trips. If you do the later, I would suggest Laminai as it gives you a nice boat ride down the river and ends up at some nice ruins.
No matter what you do, it will be fun and relaxing as long as you have a decent place to stay.

One thing I forgot to mention is the beach. The beach in Placencia is the best. There's not alot of "beach" on the cayes. With a 3 and 4 year old, Placencia may be the better choice for you.

december is a busy time all over belize. i'm partial to placencia. the best thing to do is go to all the local websites, such as placencia.com, gocayecaulker, etc. if you're going to travel some, it think the inland villages are better than the cayes....easier access.