Did you buy that snail? 'Cause seriously, I think they ripped you off. :( If you buy a plant or a fish at Petsmart or Petco, chances are you'll end up with a pond snail or two also. The little things multiply like nuts so they tend to "ride" along when you buy anything from a Petco or Petsmart tank and you don't know it until you see them crawling up the glass of your tank. I could be wrong and for your sake (and wallet) I hope I am, but usually the mystery snails they sell are quite large with solid yellow or black shells. That, unfortunately, looks like a pond snail I picked out of my tank and disposed of a few weeks ago. But truly, I hope I'm wrong, that your Petco just does things differently than mine.
Hopefully someone else will come along who can confirm or deny this.
Oh, and welcome to the forum. Glad to have you.
You're off to a great start with your betta bud in a heated, filtered tank.