The above answers are correct, but they fail to point out one key fact: Nobody actually says "Farewell" any more. You will come across people saying "Farewell" in novels, plays, songs and poems written a hundred or more years ago. You will NEVER hear this in modern English.

Farewell is as a rule used for saying goodbye to someone when you do not expect to see them again for a long time.

Goodbye is used as an everyday term usually when two people are ending their interaction (or it maybe used as a short time) while farewell is more formal and usually implies that you will never see the person again (or it will be a long time).

I think now you can better understand the difference from my answer.