Vanish and Banish, any difference?

According to the dictionary the meanings of Vanish and Banish are the same or almost the same. Is there any subtle difference?

Banish:
to expel from or relegate to a country or place by authoritative decree; condemn to exile:

 He was banished to Devil's Island.
to compel to depart; send, drive, or put away: to banish sorrow.

Vanish:
to disappear from sight, especially quickly; become invisible: 

The frost vanished when the sun came out.
to go away, especially furtively or mysteriously; disappear by quick departure: The thief vanished in the night.