Drier is a comparative adjective meaning more dry. A dryer is one of many types of electrical appliances used to dry things.
The words were once interchangeable. The distinction crept into the language through the 20th century and has only recently solidified. Some dictionaries still list the words as variants of each other, but the words are almost always kept separate in 21st-century publications.
The two words are often (understandably) mixed up—for example:
As Sherrod sat under the hair drier, a customer came and hugged her and asked for an autograph. [Atlanta Journal-Constitution].
Another sign of winter in the Fredericksburg region is much dryer air as represented by the dew point readings. [Fredericksburg.com]
These are not serious errors, but they are out of step with the prevailing usage.
These writers use the words in the ways that are now conventional:
Readings in the upper 80s to near 90 this afternoon are combining with drier air to make it a rather pleasant day out there. [Washington Post Capital Weather Gang]
The sleek Xlerator hand dryers may blast the water away, but their decibel level can spike into the mid-90s. [Chicago Tribune]
I love ageing, despite the drawbacks – thinner, drier skin. [Guardian]
It was “well equipped” with all modern conveniences, including a combination washer-dryer machine. [Telegraph]