The Seiko Museum Ginza gives the etymology: "Combining the Greek words 'chronos', meaning 'time', and 'graphos', meaning 'to write' or 'to record', the name chronograph literally means 'a device to record time'."
The distinction that matters. A chronograph is a function: the watch can start, stop and reset a measurement of elapsed time. A chronometer is a rating: a designation whose definition, test programme and minimum requirements are laid down in ISO 3159 and awarded after testing by a neutral body.
The two are independent. A chronograph may or may not be a chronometer; a chronometer may or may not be a chronograph. The museum notes the confusion itself: "A similar word is 'chronometer', which refers to a device used to express time extremely accurately."