@anamur-- I think comets' tails do become smaller and smaller and become nonexistent as they get farthest away from the sun.
I didn't read this anywhere but my teacher showed us pictures of Halley's comet when it was first seen. Astronomers took pictures of the comet for 14 months and the pictures showed the tail getting bigger and stronger as it neared the sun. The tail got smaller and smaller as it went farther away from the sun. In the last picture, the tail was like a small white dot.
So I guess the tails reform afresh every time their orbit nears the sun. And when they're in the "snow line," they pretty much die out as you said.
I think comets are the most fun to learn about. It's so cool that they're tail always points away from the sun. And it looks kind of curved too, it's not like a straight line.