Finish + object
Finish is used to talk about getting to the end of something or completing an activity.
- Have you finished reading that novel?
- You never let me finish a sentence.
Note that finish can be followed by an -ing form.
End + object
End is used to talk about stopping or breaking something off.
- I have decided to end our affair.
End cannot be followed by an -ing form.
- I decided to stop seeing her. (NOT I decided to end seeing her.)