fdmaxey's answer is excellent.
But yes, "enjoying the limelight" means they like having everyone's attention, it makes them feel special.
Show off - is usually a term used by somewhat jealous people, or it's used in a none flattering way (most of the time).
For example, imagine there were 13 friends or colleagues in a room, and one could speak Spanish, and the other 12 would like to but could not, when the Spanish speaker showed everyone how good they were at speaking Spanish, everyone else would think that one person was a "show off".
English people tend to think that when someone is better than them at something, they should keep it to themselves hahaha