The Horrortale Snowman is the same you encouter at Undertale. He is damaged by Frisk's selfish actions.
In the comic, Aliza will rebuild him again, and he will reward her with his carrot, and a piece of his snow.
Frisk-They were the one who made the Snowman suffer.
Aliza-At first, she wanted to walk away, but her heart told her to help him. The Snowman was very grateful, and gave her his nose and some of his snow.