Thinking about both of these tools and in reality they are the exact same tool. Google classroom basically goes directly to the quiz creation tool found in Forms, so why is one preferred over the other?
Yes while in Classroom you have the ease of going to the quiz creation function without going out into another application but therein lies my reasoning as to why I prefer making a quiz in forms.
The structure of where your data is stored in drive while using classroom is under a few layers. Every class you create in classroom drive creates a corresponding folder. Every time you create a quiz while in that classroom all the work and files are stored there. - I propose creating an assessment folder in your drive for each of your subjects and levels and then using forms create your quiz and store in your assessment folder. From the classroom you can assign the quiz in your assessment folder. Now you can also share your assessment folder with team members without them having access to your specific classroom(s) folder.
