I believe BARnifum is the registration name while Niso Fumagalli is the trade name. All modern roses have a registration name (usually indicating the breeder or hybridizer) which is used for registration with various rose societies around the globe. The trade name is the name under which the rose is sold/traded in commerce. Both names are valid.

Here's another example: The commercial name for one of David Austin's roses is "William Shakespeare 2000" while the registration name is "AUSromeo".