Re-manufactured makes it sound more official and within original specifications with parts to keep it original......... re-built (today) to me means done by an individual.
I can re-build an engine.......
I really doubt there is much difference but, you can usually get a warranty with a re-manned motor
reman usually means back to original specifications and with a warranty and usually from a large company.
rebuilt means... well a lot of things, back to stock or bored out/ special rods/pistons/rings etc. could be better or could be worse. if you have faith in the one doing the rebuilding you can spec it out the way you want rather then just getting a stock motor.
Like others have said a remand engine is like new, A rebuilt engine is new bearings, rings and gaskets. The thing to watch out for is reconditioned engines. There they only replace the worn/broken parts and nothing else.