The snow/rain gets reused when it hits the floor.
I wouldn't recommend letting it pile up and stay on screen because you would end up with thousands of objects on screen which would cause your computer to get slower and slower until it froze.
My recomendation would be to use a flat plain and draw a snowflake on it with transparent backgound to make it look more real.
[UPDATE]It's worth looking at this code if you want a non shader bullet tracer effect.