If you say "Notate the date for the event", I will note the error in it. Use notate if you are referring to notations like music, ballet, choreography etc. If you are not using a special notation, you are most likely noting and not notating.
Notate means to put down in form of writing music compositions, choreography or texts. It can also mean to put into notation.
"Unlike in the old days, you can now notate as much as an entire ballet."
Note could mean a melody, tune or song or a record of something written to aid remembrance for future purposes. Also, it is a short and informal letter.
"A. I sent him a note to remind him of our board meeting."
"B. I enjoyed the presentation, particularly the delivery of each note and key."