Do these words have a different meaning?
Should we say artificial sugar or sweetener? Should we say artificial fur or faux fur?
Is there a rule that defines the border for artificial/faux/fake?
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AHD treats them as synonyms:
adj. artificial or imitation; fake: faux pearls.
However, they are very rarely interchangeable, as Kris points out.
Faux has the connotation of 'accurate (more or less) representation / imitation with no moral intention to deceive'; it is a rather 'fancy' term (probably due to the obvious French derivation) useful to advertisers:
When manufacturing faux objects or materials, an attempt is often made to create products which will resemble the imitated items as closely as possible. However, some products are intentionally made to look "faux", for example, faux furs made for prospective buyers who want their fur to be recognizable as imitation due to controversy over the use and manufacture of real animal furs. [Wikipedia]
Artificial is the 'unmarked' term (artificial limb; artificial intelligence; artificial sweetner; artificial light) but can connote a lacking of a desirable quality ((of a person or a person's behavior) insincere or affected [Google]).
Fake has shady / tacky connotations (a fake Rembrandt; a fake gun).