Hello and congratulations,
I am also in the process of figuring out the differences and making a decision. I know you can get your letter of affiliation from a university professor for DAAD, but it would be worth looking into whether or not tuition (though minimal in Germany) is covered if you would like to enroll in classes.
I think classes also largely depend on where you are in your studies and how they benefit your research interests and project. It might be worth asking a current grantee how the DAAD grant works if you would like to enroll in classes. Did you propose a university affiliation in your grant statement?