I had a similar problem when replacing the cracked screen of an iphone4s a couple years back.
Original iPhone 4s screen glass was cracked so I ordered a replacement screen/digitizer. The replacement digitizer would not function fully (dead areas), so I got a third and had the same issue.
Turns out the original screen had a connector with a couple of slightly bent pins. When the iPhone was originally assembled, the bent pins on the screen connector also bent and spread apart the corresponding sockets on the motherboard connector. The original screen made good contact with the sockets so it worked fine.
But the replacements (with no bent pins) did not make full contact and so they didn't work.
After pulling my hair out and questioning my existence, I did finally spot the bent area of the original screen's connector (it was much easier to spot here than in the motherboard connector.
I got a small needle, and, because I'm very nearsighted, whipped off my glasses to attain a sub-atomic view of the connector. Then I tweaked the motherboard socket contacts back into position with the needle. Voilla, the new screen worked fine! Then I went back to my regular day of burning ants outside with my coke-bottle-thick glasses. Of course, ordinary mortals will need to use a magnifying glass.