I'm using a Stinger in MW3, and I see that it can level up like other weapons. Why do I care? It doesn't seem there are any attachments of any kind.
Edited for the correct answer : The Stinger's max level is 10, and as @rdurand says, you only get titles. (Sorry about the mistake, havent been playing in a long time :P)
The good thing about the Stinger is that it has a faster Lock-On than the Javelin, and usually most air streaks get downed in 1 shot with the Stinger (unless they have flares, or are hugeass copters/planes).
Stingers also have more ammo than Javelins.