"fear" is a verb while "afraid" is an adjective in your sentences. they will both have an equal meaning when written like this: "don't fear" = "don't be afraid" :)

As mentioned above beside the following fact: "Don't fear...." is not a complete sentence. You have to add an object, like don't fear something or someone. "Don't be afraid." could be a complete sentence or could be extended by adding a preposition "of" for example and a succeeding noun ( something or someone). "Don't be afraid of ....." In this case it equals the sentence "Don't fear ......."

As Romulus said. :) Because "to fear" is the verb, "don't fear" makes the phrase active. "Don't be afraid" is a state of being (ie. "be ----"), so the feeling is static.