To me logging keeps track of all states in a process, it has several levels: DEBUG, INFO, WARNING and ERROR.
But to me monitoring is very specific: collecting metrics about the state and performance of the process, to aggregate and analyze.
Monitoring is to understand how the systems behaves, and we are usually not looking at logs until something is off.
My usecase is this: I have several processes that run every day on the google infrastructure. They use new data every day. I want to keep track of the test error, and row counts for every run without going through the logs every day.