Well, we have been to both islands, so here goes"¦
Jamaica
Fairly large island (for the Caribbean) with lovely beaches, real West Indian vibe and lot's to see and do. Longer flight than Barbados, but not that much. Main resorts are Montego Bay, Negril and Ocho Rios, the latter now has a new connecting highway, so I believe transfers are quicker than the 2 hours of old, plus I think the airport has been much improved recently. Plenty of hotel choice and generally cheaper than the Eastern Caribbean islands. We've done both room only and all-inc, and although we found plenty of places to eat in Ocho Rios, we preferred all-inc for reasons stated below. Things to see include Dunns River Falls, rafting on the various rivers, croc parks, Rose Hall, Goldeneye etc. etc., though with an 18 month old you'll probably be staying put !!
I'd say the main downside is the hassle you can receive when walking out of the hotel complex or onto the public beaches, which I've always found to be far more full-on than the other islands. People try to sell you everything from jewellery, carvings and hair braiding to "coke, smoke or woman", as I was memorably offered several times whilst strolling through Ocho Rios"¦.with my wife !! This can be quite off putting and a little intimidating, especially if it's your first visit to the region.
Barbados
I can't think of a better island for a Caribbean first timer to visit. The flight is not too long and once you arrive the transfers are quite quick and the re-vamped airport is pretty swish. The whole place is very clean, safe and civilised, I don't actually think you need any inoculations and the water tastes better than the UK !! Two main tourist areas are the west coast and south coast. West coast has some very fancy hotels such as the Sandy Lane and the sea is calmer, but the beaches can be a little narrow in places due to erosion. South has more reasonably priced accommodation and restaurants, better beaches, but rougher sea, as the Caribbean meets the Atlantic. Fair bit to see and do, Harrison Caves, wildlife park, Bridgetown and a submarine, see Mark's excellent guide for more info. The people are very friendly and we experienced no hassle whatsoever, even when we spent a few hours wandering round the busy capital.
Downsides"¦well it's not cheap, though there are relative bargains to be had in November and the first 2-3 weeks of December. Some people think it a little overdeveloped, with a subsequent loss of character that some of the other islands may have, but I think it has real charm and the people are fabulous. Definitely the one I'd go for, if funds permit.