Here's a comparison among the Heathrow Connect, Heathrow Express, and the Underground for:
- approximate elapsed time, and
- convenience (subjectively judged).
Fare (as of December, 2005, assuming adult, one way travel at peak times purchased from a ticket machine or window):
Underground to Piccadilly Circus £3.80
Underground Zones 1-6 Travelcard (allowing travel from Heathrow to town as well as unlimited Underground and bus rides for the rest of the day) £12.00
Heathrow Connect to Paddington £9.50
Heathrow Express to Paddington £14.00
(note: there are some discounts on fares that are worth looking into, especially for adults travelling with children, travel during off-peak periods after 9:30AM or weekends, and so on. The details are at the Transport for London site.)
Elapsed Time (assuming starting from Terminals 1, 2 0r 3):
Underground to Piccadilly Circus 47 minutes, every 15 minutes
Heathrow Connect to Paddington 26 minutes, every 30 minutes
Heathrow Express to Paddington 15 minutes, every 15 minutes
Underground can be crowded with lots of baggage in the aisles. It's not uncommon to have to stand all the way into London. Once in London, there can be lots of stairs to negotiate with your baggage.
Heathrow Express has places in the cars for baggage near the doors.
Heathrow Connect: I've not taken this service, so I can't comment. The Amiable Heretic's post "360203 at London Paddington," has a comment on the cars themselves with a photo. The additional maximum waiting time, 30 minutes versus 15, is a factor.
[update: I've taken this service once from LHR to Paddington. It's a good, cheaper, slower alternative to Heathrow Express.]
The bottom line is that, especially during non-peak periods, it may be cheapest overall to take the Underground, especially if you expect to travel by bus or Underground for the rest of that day or you're travelling with up to 4 children who are 5-15, inclusive. Hauling lots of baggage up the stairs in the Underground can be a major pain, however.