Check your screen flex cable, does it have 3 exposed solder points? This section needs to be covered fully otherwise it can short against the metal insides of the phone. If the points are exposed then you have most likely shorted out a component on the logic board.
In the past I have seen this happen with iPhone 5 screens (all though the opposite would happen where you get display but no backlight). Failing that, re-install the old screen back into the phone (or partially) to diagnose if your issue is with the screen you purchased or a logic board issue. If you have a damaged logic board then it's extremely hard to diagnose what component could be the issue and usually even harder to fix (usually requires some super fine soldering or re-balling). Hope this helps.
eman05 could be right, it's also easy to knock components off the board where the connector for the screen is, since it's covered by black tape, usually goes un-noticed.