Use referee for football, rugby and boxing, and umpire for cricket, tennis and baseball. Both mean an official who enforces the rules, but the two words have simply settled on different sports by convention. Referee also has a second life off the pitch: a referee can be someone who gives you a reference for a job.
The core difference
The two words describe the same job of enforcing rules, but tradition assigns referee to some sports and umpire to others.
- referee — the official who runs a football, rugby or boxing match, and enforces its rules: the referee blew the whistle and waved play on; also a person who vouches for you, as in a job application.
- umpire — the official who oversees a match such as cricket, tennis or baseball: fans booed after the umpire's decision went against the home side; can also mean a neutral party asked to settle any dispute.
How to tell them apart
There is no rule of grammar at work here, only sporting tradition. Football, rugby and boxing have a referee; cricket, tennis and baseball have an umpire. If you are not sure which word a sport uses, that is worth checking rather than guessing, since swapping them sounds noticeably wrong to a fan of that sport.
Referee has a second, everyday meaning that has nothing to do with sport: on a job application, a referee is someone who can vouch for your work. Umpire keeps closer to its officiating sense, though it can stretch to any neutral third party settling an argument, sporting or not.