Well... This is a toughie. I'm not sure where to start...

In a nutshell, anything Summoner can do other jobs generally do better and/or faster. We can do a little bit of everything - damage, buffs, debuffs, and cures - but we don't really do any of it spectacularly. Back in the early days of the game, this wasn't such a big deal, since we could often be used to fill in whatever gaps there happened to be in an alliance. However, now that it's reasonable to assume that most players have multiple level 99 jobs, and can just switch around to whatever specialist is actually needed, Summoner finds itself outclassed in many events with very few niches of their own to occupy.

As a result, Summoner is not often asked to many events. The two places we really shine are in Wildskeeper Reives and Walk of Echos. In those two places, our ability to act via proxy (i.e. all our effects happen via our pets and not our characters) allow us to survive where others would die. Even so, there needs to be a lot of us there, as the tactics involved tend to mostly turn into tossing endless pets at whatever needs to be made dead.

As for a direct comparison to BST? We're fairly even in most things. BST gets a pet with more HP and fantastic DoT, but SMN gets the ability to call out pets at will and with fantastic spike damage. Our best strengths are definitely in tackling mobs that do primarily one or two types of elemental damage. There's a thread over here that shows off some of the things we can take on in solo situations.

Now, with everything else having been said... Summoner is still my hands-down favorite job. I've discovered that as the size of the group decreases, the usefulness of a Summoner increases (again, going back to the idea that SMN can fill in missing spots very well, due to the breadth of our abilities). I may not get to take it to pick-up Delve runs, but it's always my first go-to when I'm doing things with my friends. Also, I'll put up our merited Rage pacts as some of the flashiest looking abilities in the game. But best of all? No other job is tied as closely to the lore of what happens in Vana'diel as Summoner is. I'm a sucker for a good story, and I love the way avatar related things are sprinkled liberally throughout the game.