A charm is specifically applied to an object or being. Such as the Fidelius Charm(Invisibility used on Harry in book 5) or Warnd-lgihting Charm(Lumos).
An enchantment is something that can be applied to anything, even "thin air". And what I mean by thin air is the Protego Totalum(Book 7, used by Hermione at several campsites) and Priori Incantatum(Harry vs Voldemort at the end of Book 4). Additionally, charms are a subset of enchantments.
An analogy to biological classification: If a charm is equal to a Phylum, then an Enchantment is equal to a Kingdom.
It should also be noted there are more types of enchantments than just charms. Like Repello Inimicum(disintegrates objects that pass a certain threshold) would not fall under a charm because it doesn't require a physical presence(object or being) to have the spell take affect. Instead Repello Inimicum is a field around an area.