If you make a note of (or about) something, you write down just one item of information, possibly two. If you make a note of your class starting time, you just write down one piece of information.
If you take notes during your class, you are writing many short pieces of information, all related to the teacher's lecture.
"I took a lot of notes for the math class."
"She took notes during the meeting."
If you take note of something, it means that you notice it and give it special attention. It may or may not have some significance.
"He took note of the fact that she was no longer wearing her wedding ring."