Only reinforcement is correct when speaking of structures. We hired him to add some reinforcement to the frame of the building. (Not backup) When speaking of amounts or supplies, you can only use ‘backup’ I always carry a backup battery for my phone when I go hiking because I use it for navigation. (Not ‘reinforcement’) I stock some some supplies in my basement for backup. (Backup is there in case I run out)

In the noun forms, they both mean extra support, usually in a conflict or emergency situation. Reinforcment is more formal, and backup is a bit more casual.