[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