Let's start with Nymphs; they tend to be the simplest. Nymphs are, generally speaking, female spirits or minor goddesses of features, landmarks or locales. (On other words, most are linked to places or things.) Some, however were the daughters of minor gods, like Daphne. Many of them were once the local mother goddess, and were "demoted" once the main Hellenistic deities moved in, becoming the many flings of Zeus, other gods or Heroes (while others became mortal women). Some were considered the spiritual "mothers" of towns and villages, and the prominent families would claim descent from a Hero and a local Nymph, in much the same way that in many villages in Africa, the chiefs would claim descent from local gods (or Christ if missionaries had converted the local populace).
Satyrs/Sileni, on the other hand, are also nature spirits, but they are more associated with all wild places and the wildness within mankind. You would never expect to find a Satyr for a grove of trees, which would be a perfectly logical niche for a Nymph. Original depictions of Satyrs were dwarfish men with horse's tails and pointed ears. Bellingham's An Introduction to Greek Mythology uses this description, likely in keeping with the fact that hoofed depictions of Satyrs seem non-existent in the actual archeological record.
Hamilton, in Mythology, describes the Sileni as being similar to Satyrs, but having horses hooves and ears, rather than a goat's. The Element Encyclopedia of Magical Creatures seeks to split the difference, giving Sileni goat's hooves at the end of horse's legs. Hamilton says that Sileni are absent from legend, but appear on vases - see above. It seems that the main thing that Sileni were after was wine - they were followers of the drunken Silenus, or Dionysus directly. Other descriptions of Sileni relegate them to the juvenile form of Satyrs. So Boy : Man :: Silen : Satyr. In any event, if Silenus once had hooves, he lost them in artistic depictions, even though in Hamilton he is the son of Pan. Element describes him as "the oldest Satyr."
Both Pan and Silenus seem to have coalesced out of the entire groups of Satyrs and Sileni, although Pan does seem different from the older depictions of Satyrs.
Fauns were Roman nature spirits who occupied the same niche as the Hellenistic Satyrs, and thus became conflated. There is some speculation that this conflation is the source of the idea that Satyrs were hoofed. While books such as The Complete Dictionary of Symbols tend to create a 1:1 correlation between the two pantheons, as when they say that Pan and Faunus are two names for the same deity, this isn't quite the case. Faunus, according to Element, is more of a protector of the wilderness than Pan, and less interested in drinking and sex. The Romans tended to re-imagine their gods to fit the niches that already existed in Hellenic tradition, but had a number of others besides. Case in point would be Fauna. While some works list her as the wife of Faunus, others reject this - turning to Hamilton again, she is listed as the wife of Vulcan, pushing Venus out of Aphrodite's old role. Of course, if Fauna is a female counterpart to Faunus, and Faunus was the head Faun, it stands to reason that the Romans would not have had difficulty with the idea of female Fauns, as Lord_Fellgrin's mention of Apuleius' plays would indicate.
RZ-007lc. Side of Republic.