Drier is the comparative of the adjective dry. So, today it didn’t rain very much, so it was drier than yesterday, when it poured with rain all day. This white wine is drier than that white wine. Liz’s laundry has been on the washing line in her North-facing garden all day and is no drier than when she put it out. You get the idea.
A dryer is a device which dries things. So you have a tumble dryer or a hair dryer. Simple.
But: drier is also an alternative or variant spelling of dryer. How disappointing. But we don’t need to use it as such, do we? Let’s keep those two alternatives going strong, and get our laundry drier by using our tumble dryer!