"Permit" as a noun means an official written statement or certificate that gives you the right or permission to do something.
Examples: a fishing permit (= a piece of paper that you need to go fishing); a residence permit (= a document that gives you the right to stay in a foreign country); if you're an American, you need a work permit to work in Europe.
Permission is being allowed to do something. Example: You need to get permission to photograph here.
So, permit is a piece of paper and it's countable. Permission is a concept and it's uncountable.
Note that the usual pronunciation for "permit" as a noun is "PERmit" ( stress on the first syllable, unlike the verb "perMIT"), though -as usual- North Americans pronounce it whatever way they like: PERmit, perMIT, (as for teenage girls,maybe even pearmit, parmit, pormit, promit, promiscuous and so on) :P