Satyrs come in quite an array of appearance. The most common being human (typically male, female satyrs were an invention of poets around from the head to the waist, with the legs and feet of a goat. They usually also sport a set of goat or ram horns, and the tails of a goat, though some have tails that look almost like a lion or a unicorn. Some also sport the ears of a goat in lieu of human ears. It's rare, some other Satyrs have the feet and tail or a horse as well. And still some even have the facial aspects of a goat and only resemble their human half in the fact that they are biped and have hands.
Satyrs were the companions of Pan and Dionysus, who himself is also a satyr. Common to Greek AND Roman mythology, it's widely known that they were lovers of pipe playing, wine, and women. They are most commonly associated with lust and sex, and are more often than not depicted with full erections.
Fauns come from the Latin word Faunus, and it's this Latin word that introduce the confusion and the lore that Satyrs and Fauns are the same. Romans used the Latin word when speaking of Satyrs. The faun while similar to the satyr is NOT a goat, and not solely found in Greek Mythology, they are also found in Celtic mythology.
Fauns, like Satyrs, are half human half animal. Though in the Fauns case, the animal half is a deer, gazelle, antelope, ibex; it depends on where the lore is coming from. They have antlers, sometimes full branched racks, other times small nubs. The Fauns with the small nubs is another place where the confusion between Satyrs and Faun come from. Young goats, as well as young deer, have small nubs as horns and thus in cases like this, it's imperative to pay attention to the lower extremities. Fauns have the legs and the tails of deer, not goat.
In Greek Mythology, Fauns were the companion of Artemis. They were protected by her as were all animals in the forests. Some followed Pan and Dionysus as well since like their Satyr cousins, they were lecherous. However, the major difference between Faun and Satyrs was in the fact that while Satyrs were only male, Fauns came in female too.
In Celtic Mythology, Fauns were the companions of Cernunnos and Freya. Like the Greek God Pan was goat man, the Celtic God Cernunnos was a deer man usually depicted with a beautiful full antler rack.
Both Satyrs and Fauns are lust driven creatures who enjoyed a good chalice of wine and the company of both men and women.
I've found the easiest way to differentiate and to remember, is that Faun is pronounce FAWN.. and a fawn is a baby deer >~.^<