What's the difference between -galactosidase and Beta -galactosidase?

Galactosidase does not exist. It is either alpha- Galactosidase  or beta-Galactosidase. Beta-Galactosidase cleaves the beta linkages in glycosidic bonds between galactose and its organic moiety while as alpha-Galactosidase cleaves the terminal galactosyl moieties from glycolipids and glycoproteins.