I don't want to create a false dilemma here. You can and should likely use both tools in your classroom. It's really more of a question of determining your purpose and being intentional in your instructional choices.
That said, there are advantages and disadvantages for both tools. This blog entry will break down how each tool could be used and the pros/cons.
(Note that although Google Classroom is a tool available for Google Apps for Education users, I am simplifying the discussion by omitting it here, mainly because the use of the enterprise edition of Schoology is what our district supports and encourages.)
THE GOOGLE OPTION:
THE SCHOOLOGY OPTION:
Last but not least, don't forget that as part of your Materials in Schoology, you can "Add Media Album." This is the only way to create a place in your Course where -- if enabled in the advanced settings -- you can have both students and teachers contribute media to a shared space. For more directions and information, visit this Schoology support page.
Do you have other ways to electronically share or distribute material? Post your thoughts in the Comments below.