First, if you're a Clydesdale then you should probably have a triple.
But overall, the advantages of a triple are:
A triple can spin up an elevator shaft.
Gear changes are more natural and accurate.
Your legs will spin out before the top gear does.
In practice, a triple has a better chainline.
A triple is kinder to your knees.
The advantages of a double are:
A few less ounces and a few less dollars.
A narrower pedal stance, 7mm.
On paper, a double has a better chainline.
A double is a leetle more aero.
The weight cost of a triple comes down to:
Inner chain ring + chain ring bolts, (Chorus, 767 - 670 = 97g)
Long cage rear derailleur vs short cage, (Record, 205-187 = 18g)
Triple front derailleur vs double, (Centaur, 107-92 = 15g)
Wider bottom bracket, 115 vs 108, (Phil Wood, 221-208 = 13g)
Actually, it's less than the 97+18+15+13 = 143 grams total. The 97g compares a 175 vs 172.5 because that's the data WeightWeanies had available.
This is a very good history of chainring sizes.
I'd add that Campagnolo reintroduced the triple into their 10S and 11S lines.
Pro riders use race and compact doubles, especially after Tyler Hamilon's heroic performance in the 2003 TDF. If you're sponsored and you want a 6800 gram bike, maybe you don't want a triple. But you only have to spin past someone standing up once to get the point of a triple.
BTW, the 11S exists mostly to hide some of the shortcomings of the double. For me 3x10S, 53/42/30 x 13-29 covers everything everywhere ever. I don't see the point of 3x11S.
Well, it should be pointed out that Campy etc now have RDs that can handle up to 32T. A compact's 34-32 is comparable to a triple's 30-29. Sheldon Brown points out that you can go lower than a 30 on a triple by swapping out the inner, down to about a 26. But this is pointless for most people and for myself, I'm happy with my triple.