die Schnauze
snout
The front part of the head in certain animals, consisting of the nose and mouth.
Der Hund stupste mich mit seiner feuchten Schnauze an.
The dog nudged me with its wet snout.

snout / mouth
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snout
The front part of the head in certain animals, consisting of the nose and mouth.
Der Hund stupste mich mit seiner feuchten Schnauze an.
The dog nudged me with its wet snout.

Careful: usage is highly dependent on context! Only the meaning 'Tiermaul' (animal muzzle/snout) is neutral. All other meanings are colloquial, and using it for the human mouth is very rude.
Der Tierarzt untersuchte die Schnauze des Bären.
The veterinarian examined the bear's snout.
The meaning of 'Schnauze' (snout/mouth) becomes clear from the subject or object. With animals, it means the muzzle or snout; with vehicles, the front; and with people, it is a colloquial or insulting expression.
Die lange Schnauze des Rennwagens verbessert die Aerodynamik.
The long nose of the race car improves the aerodynamics.
The word 'Schnauze' (snout) is used for the mouth-and-nose area of many mammals, especially dogs, foxes, bears, or pigs.
Der Fuchs steckte seine neugierige Schnauze aus dem Gebüsch.
The fox stuck its curious snout out of the bushes.
The expression 'die Schnauze voll haben' (to be fed up) is a fixed unit. It is often used with the preposition 'von' and the dative to express what you have had enough of.
Ich habe die Schnauze voll von deinen ständigen Ausreden.
I'm fed up with your constant excuses.
The command 'Halt die Schnauze!' (shut your mouth!) is a very direct and rude way to tell someone to be quiet. It should only be used in very informal or emotional situations.
Sei still! Halt einfach die Schnauze!
Be quiet! Just shut your mouth!
mouth
'Mund' and 'Schnauze' can both refer to a part of the face, but they fit very different situations. 'Mund' is the normal word for people and is generally neutral. 'Schnauze' refers to the front part of an animal's head, with the nose and mouth. For the human mouth, 'Schnauze' is crude and often insulting, especially in commands to be quiet. Learners may confuse the words because both can be connected with the face and with speaking. For people, 'Mund' is almost always the safe choice.

Der Hund steckte seine Schnauze neugierig in die Tasche.
The dog curiously stuck its snout into the bag.
Das Kind hatte nach dem Essen Schokolade am Mund.
After eating, the child had chocolate around his mouth.