You’ll automatically skip most of the puzzles. Poor Papyrus.
•Rather than getting one Snowman Piece from the snowman, you’ll steal three. Ouch.
•Snowdin Town is empty when you arrive. You can steal 758 gold from the item shop, not that you’ll get much use from it in the future, and as many items as you need. Steal a bunch of Cinnamon Buns for future use. All of the other buildings will be devoid of life. (There’s one person left, mainly because he’s necessary for the next area.)
•It's possible that the save message telling you how many monsters are left will change to 'That comedian...' while you're proceeding through Snowdin. I'm not entirely sure what triggers it - I think it appeared after a fought Glyde, or at least after speaking to Sans near its cave? - but it will prevent you from seeing how many kills are necessary to clear the area. This does not mean your run is over! Just keep going until Snowdin is clear and you get the terribly-depressing Genocide music.