[MisterMashable]

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.

[jwmerrill]

There's a bit of discussion of how differential forms fit into the geometric calculus framework in [1], which is probably the most concise and readable introduction to the geometric calculus approach to differential geometry. All the machinery of forms is available as part of geometric calculus.

Geometric algebra/calculus has a more direct way to deal with metrical information using the dot product. Forms only use the wedge product: in problems where the dot product would be useful, forms simulate it by applying the hodge dual twice, which is a less intuitive and less direct way to get the job done.

[1] The Shape of Differential Geometry in Geometric Calculus, see section 19.4 for the bit about forms http://geocalc.clas.asu.edu/pdf/Shape%20in%20GC-2012.pdf