I think the differences are to do with American English and English. People from the USA say "to brush your teeth" People from England say "to clean your teeth" The meanings however generally are the same - unless maybe you are an orthodontist :))

Generally speaking, no. Either action will have the same result. If you brush your teeth, they should become clean. If you clean your teeth, usually it is from brushing them. "Cleaning your teeth" however may involve more than just brushing them, the person may floss their teeth, rinse with a mouth wash or practice other ways to clean their teeth. Hope this helps you :)

''To brush one's teeth'' refers to the ritual we do every night before we to to bed. ''To clean one's teeth" refers to our annual visit to the dentist's office to have the plaque, tartar, stain etc removed from out teeth. This job is usually done by a dental technician, not by the dentist.

I have never come across "clean teeth", it is possible to say: teeth cleaning, and "cleaning" is a noun in this case.

They are different, because the brush is not the only tool for cleaning teeth :-) But when you brush, you can use whichever you want and both will be adequate.