As previously Christina Braunsdorf mentioned you its same antibiotic, only difference is Gentamicin sold by different brand names...one of them is Gentamycin.
The differences between gentamicin and gentamycin is the source that taken for drug formation , gentamicin is synthetic type while gentamycin is natural source.
Gentamicin is the correct spelling for the antibiotics. gentamycin is an incorrect spelling in the literautre and has also been used by some manufacturers. The reason as explained by [Ref1] "gentamicin does not come from the genus Streptomyces but from the genus Micromonospora. It was decided that only those aminoglycosides derived from the genus Streptomyces, which includes streptomycin, neomycin, paromomycin, and kanamycin, should be spelled with a "y." Since gentamicin is derived from a different genus, it is not spelled with a "y.""
There are many synonyms for Gentamicin, including Garamycin Gentavet Gentamicins Gentacycol Cidomycin Gentamicinum etc.