Is there a difference? Both versions are common.
If there is a difference, which do I use when, and why?
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Grammatically they are interchangeable. The very definition of "unavailable" is "not available". However, my gut tells me something else. When I see 'unavailable' it immediately connotes that it's a temporary condition:
The senator was unavailable for comment. He may be available later.
The web page was unavailable all afternoon. But then it came back online.
Whereas when I see 'not available' it implies to me that it's a stronger or more permanent situation. Maybe because of the emphasized not?
This service is not available in New York. And it's not likely to be any time soon.
The web page is not available outside the company. Because it's private.
While you could legitimately substitute "not available" for "unavailable" in those examples (and vice-versa), I wouldn't.
Both are perfectly valid, but I would take issue with the implications of OP's "Both versions are common" - relatively speaking, they're not...
In short, unavailable is the standard form. But noting OP's comment regarding negation vs. not, I suggest that, for example...
This service is not available to our viewers in Northern Ireland.
...is normally preferred over...
**This service is unavailable to our viewers in Northern Ireland*.
Both OP's versions mean exactly the same, but as @Lynn says, the reason "un-" is more common is because it's temporary. With something more "permanent", like my example above, or the more extreme "Not available at any price", the "un-" version is actually significantly less common.
They both mean the same, they are just different constructions.
Temporarily unavailable seems "lighter", as it contains only 2 words, whereas temporarily not available contains 3, which are redundant, as unavailable is a grammatical word which bears all the required meaning.
Use less words, be easier and faster to understand.