In my opinion, Edmodo and Google Classroom are both considered to be a virtual classroom, they both serve the same purpose. They assist teachers with their classroom workflow and assist with managing classroom assignments.
Both applications will allow teachers to post assignments and announcements, however, since Edmodo has been out for a longer time, it has more features that is not included in Google Classroom.
One of the biggest advantages Edmodo has is it allows teachers to have a co-teacher account. For example, if there are two teachers teaching together in one classroom, Edmodo allows you to share that group with both teachers. In addition, it also allows you to create quizzes directly in the group, access a class calendar, and install apps from the Edmodo store. Another advantage Edmodo has is Snapshot. Snapshot is aligned with the math and language arts common core standards and it allows teachers to create assessments for these subjects in grades 3-12. Google Classroom, unfortunately does not have these features at this time, but I think Google will improve the Classroom application and have these or similar features eventually.
However, if you are using Google Apps for Education in your school district, I would highly recommend using Google Classroom. It is much easier to hand out and manage assignments using the Classroom interface. Classroom gives the teacher the option to allow students to view, edit, or make a copy of the document for each student. If the teacher hands out an assignment and selects, 'make a copy for each student', it will put a copy of the document in each student's classroom folder in their Google Drive. If the teacher selects, view or edit, it will keep one copy of the document and will allow the class to share the document. Although, Edmodo allows you to integrate your Google Drive, I don't think it handles assignments that way.
Furthermore, Google Classroom automatically creates classroom folders for both the students and the teacher in their Google Drive. There is no need to use gClassFolders to manage student rosters and folders. Once the student joins the group, Classroom automatically handles this task for the teacher. Edmodo does not interact with student folders.
Another feature Google Classroom has is it allows the teacher and students to email each other. Note, they are only allowed to email within their group. This can be a great feature for communication and feedback on assignments or if a student is absent.
Again, if your school district is using Google Apps for Education, all of the apps integrate well with Classroom. In my opinion, it just makes sense to use it!
In conclusion, I am introducing Google Classroom to many teachers in my school and some are embracing it, while others still prefer to use Edmodo, and that is okay. I certainly do not want to force an application on teachers when they do not want to use it. However, I feel it is up to the teacher to determine which application works best in their classroom. I think they are both great tools and definitely should be utilized.