@stringer is better at this...buuuuuuut:
Wire edge is what we generally think of as a burr. It's the wire of metal that forms at the edge that we are flopping back and forth then removing via one or more methods.
Foil edge is not that different, but basically a very thin "foil" at the very end of the edge, often less than a micron thick and generally not that even. It will feel sharp as hell and slash thin paper or human hair, but won't last on a board past one stroke generally: it will fold over and require you to touch up the edge.
Good deburring techniques will get rid of both, generally, as long as they are properly "worked."