You're mistaken, GA does have a Hodge star, as I've explained many times before https://grassmann.crucialflow.com/dev/algebra
The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.
The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.
In any case, you did not attempt to answer my original question. Are GA and DFs just different ways to define "equivalent" concepts or is there some more fundamental difference that I am missing?
You obviously know more than me about this, so I will ask you a slightly different question: if I learn GA well enough and totally ignore differential forms, what will I miss?
Differentials are a concept that the comes from doing calculus on manifolds, and exterior products of differentials are just used for tracking information about oriented volumes.
To answer your question (switching differentials forms for for exterior algebras), you wont miss anything, as the wedge product is part of a GA.