FBNeo takes both formats, but other Neogeo cd emulators only take the cdz patched versions. It may be better to keep only those.
Redump files are in .CUE format with several .BIN files called multibin. This core does not support multibin and you must convert your games to have a single bin file.
You can do this with the CDmage 1.02.1 software:
File > Open > Select your game in .CUE format
OK.Sources: