I try to state the differences in your example.
"Sie ist sehr eigen, was Musik betrifft." != "Sie ist eigenartig, was Musik betrifft".
Sie ist sehr eigen, was Musik betrifft. ≠ Sie ist eigenartig, was Musik betrifft.
The word "eigen" is more positive, meaning that the person is peculiar but you are affirmative of that and you accept it. Like an eccentric rock star is accepted.
Being "eigenartig" has a bad taste to it. You say that if you are not fond of the person's behaviour and you find it weird in a way and you delimit yourself from the person.