Cambridge Dictionary definitions:

To commemorate means to remember officially and give respect to a great person or event, especially by a public ceremony or by making a statue or special building:

For example:

Gathered all together in this church, we commemorate those who lost their lives in the war.

A statue has been built to commemorate the 100th anniversary of the poet’s birthday.

To celebrate means to take part in special enjoyable activities in order to show that a particular occasion is important:

For example:                                                   

We always celebrate our wedding anniversary by going out to dinner.

If this plan works, we’ll celebrate in style (= in a special way).