Pixie hair cuts and crops are both very short haircuts, and if you had to say a difference between the two, aside from the fact that "crop" sounds terrible because it's something you do to your dog, it'd be that a pixie haircut can be a little bit wispy. A crop might be more blunt while a pixie might be a little more textured and razored- and this is in the sense that these kinds of short haircuts were once associated with "cheerful" fairies (pixies).
Another term, "gamine crop"- which sounds dangerously close to "feline," "famine," "game" (as in hunted wild animals) and "gamey" (disreputable and scandalous), is supposed to describe these kinds of short haircuts as wispy and textured and slightly dishevelled (in a trendy kind of way, like boho hairstyles), and that's because "gamine" is French for a girl who hangs around on the streets. I'm not sure about you, but these associations- between hair cuts and starving feral animals or homeless delinquent waifs with grubby fingernails- don't quite make me want to get a pixie haircut or a crop.
Choose a word you like (pixie hair cuts, crop hairstyles and gamine crops pretty much all refer to very short hair), but most importantly remember this: if you're thinking of getting a new short hair cut, avoid any confusion and bring a few pictures to your salon; perhaps one you've seen here.