Themes, motivation, depth, and consistency.
Though not all my characters possess solid factors in all of these (I seriously need to rewrite some of this crap) I do believe they hold a lot of importance.
RWBY, though not needed on site, has a lot of theme based stuff canonically with their characters, even if it's not all too obvious at first. Though that doesn't mean they can't stray from it. Cookie cutter protagonists feel generic on paper so throwing something new here and there help spice things up.
Motivation plays a lot in determining how a character would turn out in the long run. What happens if their motivation is challenged? What happens if they lose that motivation? Do they lose their drive? Or do they go through the fire and temper themselves into a stronger person?
Depth. Self explanatory. Having a fighter be good because they're good is boring. Having them be good because they trained to attain a certain goal or hope to attain one is more interesting. Having a fighter train to be good because they feel self-loathing because they feel powerless in the face of all those skilled people around them and to prove they're not just comic relief in the eyes of the audience. Well that's Jaune. And that's not bad at all. On paper at least.
Consistency. At least flow-wise. Writing how cool they are in one sentence then jumping straight to how they're quiet and shy is jarring. Giving the character a certain speech pattern or tendency or a neat quirk is interesting, if they don't overdo it at least.