When you say:
His motive (for the crime he committed) remained a mystery.
It doesn't sound the same if you change "a mystery" to "mysterious" as if the two words didn't have the same meaning. Please correct me if I am wrong.
Another example is:
‘I quickly went over to the desk and starting sifting through the papers, looking for a clue to the identity of our mystery guest.’
If you Google search them, you have 718,000 hits for "mystery guest" and 142,000 for "mysterious guest". I assume there has to be a reason for that.
‘A mysterious benefactor provided the money’
In the above example, it sounds like it is OK to use "mystery" in place of "mysterious" (personal opinion). However, you get only 17,000 hits for "mystery benefactor" and 135,000 hits for "mysterious benefactor", which is 8 times more. The usage of "mysterious benefactor" seems quite differnt from "mystery guest".
[Oxford Online Dictionary]
In the first example, "a mystery" looks like it is functioning as an adjective with a different connotation from "mysterious". Is it impossible to say "His motive remained mysterious"? Is it just idiomatic (just the way it is)?
In the second and third examples, what happens if I use "mysterious" in place of "a mystery" or vice versa? Why do you think there is such a big difference in Google hits?
Please don't asnwer with "that's the way it is".