Ein Kater is German for a tomcat, but it also refers to a hangover. Anyone who's had the kind of hangover where it feels like a scraggly, smelly tomcat is clinging to your head will understand this immediately!
Muskel just means muscle, so a Muskelkater is a muscle hangover — or delayed-onset muscle soreness. I think it's plain which term is nicer; it's the one that allows you to imagine grouchy cats hanging off your arms and legs in the days after training.
This post brought to you by the fact that I've just started going back to the gym (and actually rather enjoy a light Muskelkater).