Commission has a lot of meanings. Some of these are similar because of the root word 'commit'. For example:
Commission: A very formal word, not used much. A duty that a person or group is given. Often an authority gives this duty without the choice of the person receiving it. Examples:
- a soldier's orders.
- In christian theology "The great commission" is the final job that Jesus gave his followers"
- a special committee appointed by a government to investigate something.
- Being given such a duty is called being commissioned.
A commitment is normally a more voluntary duty.
E.g. "I have commitments to work, commitments to my wife and children."
"When we got married, I made a commitment to my wife to care for her"
"Our company has made a commitment to our customers to providing good service."
Other definitions are totally different, for example commission of a crime.
Some senses of commitment imply exclusivity:
Commitment to my wife
Which TV will I buy? I must commit to one (make a commitment).