Note that there are several alloys that could be called "stainless steel", and they have different characteristics. A rifle barrel is probably a 400 martensitic series alloy (nickel-free and hardenable), a type that has good characteristics for machining, and is optimized for hardness (other applications include knives and surgical instruments). We should note that non-stainless barrels are usually made of chromoly, which is harder than ordinary steel but not corosion resistant.
The advantages to stainless is hardness and corrosion-resistance (you can corrode a stainless barrel, it's just harder to do). A harder barrel means that, all things being equal, it will wear more slowly than a chromoly barrel. Of course, all things aren't really equal, and there are a lot of factors in what could damage a barrel, but obviously a harder steel has some advantages. That said, a well-maintained chromoly barrel will probably outlast a poorly-maintained stainless barrel.
Some people say that stainless is easier to clean (probably because of the hardness). It sounds reasonable, but I think there are so many factors involved in that that there probably isn't a definitive opinion.
Hardness also increases stiffness, and a stiffer barrel will oscilate less, so the rifle will have the POI move less if you change the load -- especially if it is only changed a bit. This may be one of the reasons why stainless barrels have become popular in precision shooting. Better stiffness would also allow you to reduce the barrel size and weight.
Stainless is also easier to care for -- there is less concern that things be oiled and protected while being stored.
The disadvantages: Stainless is quite brittle compared to most steels. That means that although it is harder, when it fails it tends to fracture -- increasing the potential for catastrophic failure. This is also why you should never use a sword with a stainless blade to cut -- such swords are for display or things like iaido (where you draw and cut air). A stainless sword can literally shatter if it strikes something -- often with a spray of razor-sharp shards. Even iaido guys shy away from stainless and prefer carbon steel or coated aluminum blades.
Please note that a significant failure in a stainless barrel is probably no more likely than in a chromoly one -- it's just that the failure might be more spectacular, and possibly dangerous.
Stainless is also reported to be more temperature sensitive (internal stress due to uneven heating and a brittle metal don't play well together). Some people claim you shouldn't shoot in cold weather (below freezing). I'm not convinced by that one.
The last disadvantage is, of course, that stainless is more expensive -- the alloy itself, and the extra labour to machine it.
I personally like stainless, especially for precision rifles. I think that if the rifle is properly cared for, most of the disadvantages to stainless are easy to manage (well, other than the price

) For a hunting or camping rifle, I'd probably go chromoly.
(Note, the above is only my opinion, based on my reading in the past, not any specific test data. YMMV).