Both the words mean "not academic," but is there any difference between unacademic, and nonacademic?
Is there any phrase where one of the words should be used instead of the other?
Non-academic is a simple neutral statement of allegiance. Non-academic experience is one you gain outside of school. Non-academic license for software is different than "software for educational use only".
Unacademic is inappropriate for academic standards. A paper that calls upon results of Tarot readings as source of prediction of construction durability is unacademic. The theory of Intelligent Design is unacademic.
Both un- and non- are prefixes to the word academic to change the meaning.
According to Oxford English Dictionary (OED):
Used to express negation.
Prefixed to nouns of action, condition, or quality with the sense ‘absence or lack of’, often corresponding semantically to ‘not doing, failure to do’ (where a verb is implied by the noun, as in non-accomplishment, lack of accomplishment, failure to accomplish) or to ‘not being, failure to be’ (where an adjective is implied by the noun, as in non-activity, lack of activity, failure to be active).
In Old English the number of recorded forms in un- is very large, the prefix being freely applied with a purely negative force to several parts of speech, which may be classified as follows: (a) simple adjectives, (b) simple adverbs, (c) past participles of strong and weak verbs, (e) simple nouns.
As far as I'm concerned, there is no distinction of meaning between non-academic and un-academic.