The main benefit is weight - because the axle is now a pipe it gains strength from diameter and hence can use less metal. Bicycle frames are made of tubing rather than solid rod for the same reason.
One disadvantage is that the balls in the bearings have to be smaller because there's less space to fit them in, so all things being equal they will wear out faster. But things are not equal, and what's made it practical to have external bottom brackets is improved metallurgy and precision manufacturing. So an external BB should last as long as a similarly priced internal one. (edit) And as lantius pointed out in the comments, the balls in an external BB will be close to the same size as the ones in a cartridge BB anyway, the "smaller" comment is mostly relevant to a comparison with open bottom brackets.
Shimano had problems with the seals in their external bottom brackets that was causing a lot of premature failure but I believe those have been solved now. We've stopped seeing them coming in for early replacement, anyway.
You will also need to change your cranks to fit an external bottom bracket. Rather than a square taper external BB's all use a splined system. Which is an improvement over the square taper but there are several types. Make sure you can get a BB to match whatever cranks you buy.
They have also changed the sizing, from memory there are only a couple of lengths of external BB available rather than 10 or more. This may mean that you end up with your chainrings slightly offset from their current positions, but with a derailleur system that shouldn't be a problem. On a singlespeed it will give you a bent chainline which is bad.
Personally I don't see the point in upgrading an existing bike. If you needed new cranks and were focussed on weight it might make sense, but it's an expensive risk (things might not work the first time) for a very small gain. For the same price you could get a Phil Wood BB that will last forever, or save money by buying cheap cartridge BB's as you need them.