「働く」と「勤める」はどう違いますか？

They both mean work, but 勤める is like a specific case of 働く.

働く (はたらく)

More general of work, paid or unpaid.

毎日働くのは大変だ。
It’s tough to work every day.

彼はよく働く人だ。
He is a hardworking person.

勤める (つとめる)

Almost always used with a specific company, organization, government office, etc. Often followed by the particle に to indicate the place of employment.

私は銀行に勤めています。
I am employed at a bank.

この会社に10年勤めています。
I have been working for this company for 10 years.

So if you 勤める somewhere, you are also 働く there.