More knowledgable users will surely chime in shortly, but from what I understand, there's two sides to this: tone and playability. A bit counterintuitive maybe, but from what I understand thicker strings have more overtones, while thinner strings have more fundamental and more sustain (?).
This idea runs around a lot, but at some point, you will also read that (when you get to big strings, like B's on 5's), thicker strings are duller. So, which is it - are thicker string brighter or duller?
I use light gauge strings (40-55-70-95-130 - all about 35 pounds tension) on my 34" 5's. Compared to 45-60 etc, the strings are "twangier" - it's easier to get the strings moving more, and when strings move more, the tension shifts a bit at the initial part of the note - the note goes up a tiny bit in pitch. It's something that's somewhat subtle - it doesn't sound like you're out of tune, it just adds "energy" to the proceedings. Thicker strings don't have as much of this - if I play a bass with thick strings, it feels like I've lost access to dynamics. When I play hard, it doesn't sound like it.
Thinner strings also make it easier to do a bit of controlled fret "bounce" on the initial transients - there's more zing in the note as you dig in a bit. Given that, the way I play, thinner strings are definitely brighter sounding - they have more (not less) harmonics.
And yes, you'll have to turn up your volume control a little bit. Less metal means a little less output from your pickups.
Thinner strings are also kinder on your neck. As someone who does setups for a bunch of people, if you want a stable setup that lasts a while, thinner strings are your friend.