The Heathrow express starts at Terminal 4 and its first stop is Terminal 3 (same stop for Terminals 1 & 2) so you don't need to transfer (although the train could already be crowded). Currently there's no subway (tube) access at Terminal 4 so the problem is for people at terminal 4. Both options are available at terminal 3.

I haven't figured out the fine details, but I discovered in March that the Express trains to both Gatwick and Heathrow were 'included' in a day travelcard (virtually unlimited travel on both Underground and buses in London) on both Saturday and Sunday. I don't think that applies during the week - but possibly does off-peak. The travelcard is about half the cost of the Express!! (I know makes no sense).

And for completeness, at terminal 4 they'll sell you the travelcard on the arrivals level - at the opposite end to the train! (and they also take credit cards). At terminal 3 the travelcards are sold at the Underground station which is a fair distance (5-10 mins walk) from the Express service.

Hope this helps. Alan & Catherine