As nouns the difference between perm and curly

is that perm is a permanent while curly is a person or animal with curly hair.

As a verb perm

is to give hair a perm, using heat, chemicals etc.

As a proper noun Perm

is historical region of northeastern Russia, corresponding to the Kama basin, a left tributary of the Volga. In Russian known as Пермь, also Пермь Великая "Great Perm" or Пермь Камская Kamskaya Perm after the Kama River.

As an adjective curly is

having curls.