Tom Snout is part of a subplot of the play that concerns a group of "rude mechanicals" (description coined by Puck) who are meeting to rehearse a play that they hope to perform for Theseus and his guests on his wedding day. The most famous of this group is Bottom, who is turned into an ass by Puck.
Snout is the tinker, which we might consider today as someone who does odd jobs. Tinkers would travel around with a cart of wares and tools from house to house to sell items and fix things around the home.
His name certainly conjures up an animal's nose, but whether that is any reference to the size of the nose of the actor Shakespeare originally had in mind for the part, we cannot know. The mechanicals are meant to be the "clowns" in this play, and as such, it could be very useful for this actor to have a large nose, even if the only joke is connected to his name. Clowns often relied on physical humor.
Quince originally casts Snout as Pyramus' father, and Snout doesn't really say much in this first mechanicals scene. Once the crew moves their rehearsal to the forest in Act III, scene i, however, Snout has lots to say. He agrees with Bottom that the lion will frighten the ladies and suggests that they have a Prologue saying that the lion is not real. He suggests they rely upon the real moon for reference in their play and also that it will be impossible to bring in a wall, an important set piece in their play. He comes across as very concerned over the "reality" of the setting and characters of the play, and also ready with a solution. Bottom, whom all the other mechanicals defer to, suggests that one of their company will present the wall in the play, and though it is not decided here, that man ends up being Snout.
In Act V, in the play, he does have lines, but since those are his "wall" lines, it is not really possible to use this text to characterize Snout.
As with any clown in an early Shakespeare play, most of the things that an audience learns about Snout will come from the behavior and antics which the actors develop to increase the comedy of the scenes. These choices will vary from production to production as they are not set down in the text. So, you can learn some things about Snout (or any of the mechanicals) from reading the text, but lots more from seeing the play performed live.