I looked all over and couldn't find any difference other than maybe "Court" duty would refer to a Federal court case vs a State case. Every time I looked up "court" duty, it referred me to jury duty. I'm sure someone will come along and have the difference but I sure couldn't find one. Sorry.
I could only come up with 2 situations:
1 - Different terms for the same thing, depending on region.
2 - Sometimes being a witness could be called "court duty".