Yes, they operate on the same objects. The main operations on differential forms are the exterior derivative d and the wedge product f /\ g. These are independent of the metric. If you add a metric you get one extra operation called the Hodge star ⋆, which is a prefix operator, so it operates as ⋆f.

Alternatively, if you have a metric on your space you can define the geometric algebra derivative D and the geometric product. These are closely related to the differential forms operations, e.g. Df = df + ⋆d⋆f. Using it, Maxwell's equations become one equation, namely DF = J where F is the electromagnetic field tensor and J is the four current.

So in my opinion it's indeed best to not put this as differential forms vs geometric algebra, but rather as a single theory in which some operations don't depend on the metric (d, /\) and some do (⋆, D, geometric product).

Unfortunately, geometric algebra has been overhyped by its proponents, and the papers they write are less than rigorous, so some people are under the impression that it's crackpottery, so it's safer to call it Clifford algebra :)