The pros: it was fast, trains were frequent, reasonable easy to use, and it had a good coverage of the areas of London that we wanted to visit.
The cons: it was a bit grimy, it was crowded at rush hour, it's underground so you don't see any of London when you travel, and it's expensive. Also, when transferring between tube lines you usually had to do a fair bit of walking, using up and down stairs. Definitely not transportation for the handicapped.
If you plan to be a tourist in London, I recommend getting a Travel Card. You can get them in London, or you can get them before you go. The latter is probably easier for tourists since its one less thing to do once you get there. You can get either a 1-2 zone card, or a 1-6 zone card. Unless you expect to be traveling a lot in zones 3-6, just get the zone card.
The biggest advantage of the Travel Card is that you don't worry about using the tube; you just hop on where you are, transfer as needed, and hop off where you want to go.
You can also use the travel card on the buses. We took the bus once, hoping to see a bit more of London. It was awful: it was slow, hot, and we couldn't see anything.