Well to me the meaning of two expression would be quite the same, but I think some words don’t fit well in the expression “〜間(kan)”.
- I suspect 和語 will generally fit well with “〜の間(aida)”
- I suspect “〜間(kan)” is used rather for a fixed expression
My advice is,
If you are not sure which to use, use “〜の間（aida）”, then you will be on the safe side. I can’t come up with an example you can’t use this expression when you can use “〜間(kan)”.
(Maybe it is partly because, “kan” has a lot of
Expression with the same pronounciation, so “Aida” is preferred generally)
If you wish to make the expression shorter, use “〜間(kan)”.
Anyhow, I think you just need to know there are practically no difference in the meaning. (Although there might be some exception I can’t come up with right now)