Chosen Solution

I swapped a 2020 Macbook Pro 13” screen that had a dent with my older 2018 Macbook Pro 13” screen that was in good condition. Neither screen works on the opposite computer. Only an external display works. Before I go through the trouble of swapping them back are they not compatible with each other?

Yep! Square peg round hole problem! The displays are not swappable! The newer display offers TrueTone to alter both the camera and displays color which uses a different signaling. To add to that your 2020 system requires a used display as a virgin display wouldn’t have had its display calibrated so it would require access to the calibration tool which only Apple has access to. The still newer M series systems now require an Apple supplied display as the display is now serialized to the logic board (married)