Haircut and hairdo share one meaning but not two.
Both words mean "a way of wearing the hair" or "a way of cutting and arranging one's hair":
What do you think of that guy's haircut/hairdo?
Do you like my new hairdo/haircut?
But when you literally make your hair shorter by having it cut by scissors, only haircut is correct:
I made an appointment for a haircut on Thursday.
You really need a haircut -- your hair is getting in your eyes.