From experience I would have to say 'differential forms' seem to fit the bill for a unifying mathematical language as applied to geometry and physics. GA seems to me as more of a computation tool. Differential forms are pretty standard in many maths and physics texts. Many paper on the preprint Arxiv use differential forms. The only thing about DFs is that they are very efficient for communicating ideas and doing proofs. For practical calculations they don't really simplify anything (actually get in the way) but it's easy to transform them into the usual vector tensor notation. GA seems less flexible in this regard, you take the product it uses and you either like or lump it.