I created a snowman in my Survival world (seed: 1537751306) in x:255.7, y:32 z:293 (if you see in the game, I dug a hole in where coal vein was).
I expected the snowman to leave a trail of snow. But it did not. I killed him and created another snowman, but it also did not give off snow.
I also tried in a different world with creative mode. It still does not give off snow.
Edit: Not sure if this should happen, but it died by one shot of arrow by a skeleton and left 4 snowballs, not 8.
If there's any more information to be known, please ask me.