I'm curious as to whether "Schnute" and "Schnauze", both meaning some variation of "mouth" or "snout" are both cognate to "snout". If so, it would appear that they have been affected differently by sound shift, and I'd be interested in how this would have come about historically. I do not have access to proper etymological literature unfortunately.
Yes, they are cognate. The (openly accessible) Etymologisches Wörterbuch des Deutschen: Schnauze sais:
Die Formen [die mittelniederdeutschen Wurzeln von Schnauze und Schnute] gehören wie gleichbed. nl. snuit, mengl. snūte, engl. snout, norw. snute zu der lautmalenden, etw. Vorragendes bezeichnenden Wortgruppe mit anlautendem germ. sn-
These forms [the middle lower german roots of Schnauze and Schnute] belong together with netherlandic snuit, middle english snūte, englis snout, norwegian snute to the group of words with beginning germanic sn-, who are onomatopoetic and denote something that stands out.
While Jonathan's etymological statements are fully valid, I'd like to add some notes about their usage changes during the centuries. The word Schnute was never a widely used word for an animal's mouth, while the word Schnauze was and is. Schnute was originally commonly introduced as an equivalent of Schnauze for humans in a pejorative context. Nowadays, Schnute isn't that negatively connoted anymore in general as it was in the past.
Also worth to mention: The german "Schnauze!" for the english "Shut up!" is very common, but not "Schnute!". This emphasizes further on the change of its original negative connotation.