Mounts (horses) are rented from vendors for 5sp. They take you along a predefined path. You can jump off of them at any point by using your jump key. They are exclusively to speed up transportation and prevent you from having to deal with monsters on the way to and from where you are going.
Pets: I can only speak of an enchanter pet and other classes have vastly different types of pets (Necromancer, Animist for instance). But the enchanter pet is a summoned being that comes in the form of a tank, healer or caster. They are 88% of the players level. Depending on the spec line chosen they are anywhere from a decent tank to a weak tank. They are designed to augment the enchanter's nukes, focus shield and other spells.
I hope this helps some.
Bevitt
33rd Enchanter
Percival