A chronograph is a watch with a built-in stopwatch. A chronometer is a watch certified to keep highly accurate time. They sound alike but describe completely different things, and a single watch can be one, both, or neither.
The short answer
Chronograph means function. Chronometer means accuracy. A chronograph has pushers and sub-dials to time events. A chronometer has passed strict accuracy testing. The words get confused because they share Greek roots about time, but they measure different qualities.
What a chronograph is
A chronograph is a stopwatch complication added to a normal watch. You start, stop, and reset it with pushers, usually beside the crown. It times laps, workouts, cooking, or anything else while the watch keeps telling regular time on the main hands.
What a chronometer is
A chronometer is a watch whose movement has passed an independent accuracy test, most famously from COSC in Switzerland. To earn the title, the movement must keep time within a tight daily tolerance across temperatures and positions. It's a certification of precision, not a feature you operate.
Can a watch be both?
Yes. A watch can be a chronograph and a chronometer at the same time. It has the stopwatch function and it passed accuracy certification. Plenty of watches are neither, and some are only one. The two labels are independent.
Why the confusion happens
The names are nearly identical and both come from Greek words for time. But one describes something the watch does for you, and the other describes how well the watch keeps time. Keep graph tied to timing and meter tied to measuring accuracy and you'll never mix them up.
Which should you care about?
If you want to time things, look for a chronograph. If you want proven accuracy, look for a chronometer rating. Neither is better than the other because they aren't competing. They answer different questions about a watch.