@ck997 I'm stumped. This doesn't look like any display/LCD failure I've ever seen. If you can afford to lose the data you might try a DFU restore (google it if need be).
If that doesn't work then it's most likely a hardware problem, but I hesitate to suggest replacing the screen as I don't have much confidence that would help. That being said, however, there really isn't anything else to try. If it's not the screen then it's almost certainly a motherboard problem and those can get expensive to fix; generally board-level repairs on an older phone like yours can very quickly exceed the value of the phone.
Given that the price of LCD screens is at an all-time low, if it was mine my next step would be to try another screen. If that didn't work you could try taking it to a repair shop; they may charge you for a diagnosis but at least then you'd know what you're up against.
If you do decide to try another screen, just plug in the new screen without bothering to transfer the home button first. The less you mess with those delicate flex cables the better, so if the new screen comes up with the same snow then you haven't jeopardized the button by trying to move it.