ATMEGA328P-U vs ATMEGA328-PU
The main difference is the bit before the -. That is, the 328 vs the 328P.  The "P" there denotes "Picopower" which allows the chip to run at very low power consumptions. 

Basically the P version is a more modern version of the non-P chip.  There are probably other internal differences too but you will have to check the datasheets thoroughly for those.

Everything after the - is to do with the packaging and environmental grading of the chip.  For example PU is a -40C to +85C grade DIP chip. ANR is a -40C to +105C grade TQFP chip, etc.

If you see 328P-U then that's probably a fake chip. There's no such thing as single U suffix.