"False teeth" is used when we are talking about a partial or full set of dentures, which are removed at night. Nobody uses the term "artificial teeth" to refer to removable dentures. "Artificial teeth" would be used to refer to a manufactured tooth or teeth that are permanently implanted in the jaw bone, although usually, these are simply called "implants" (can be singular or plural).

They are the same thing, and are also known as "dentures." false teeth Definitions plural noun a denture, esp a removable complete set of artificial teeth for one or both jaws http://www.collinsdictionary.com/dictionary/english/false-teeth