>Geometric algebra is, as the article points out, a more powerful version of the usual vector notation
That's not just a gross oversimplification, this is also flat out wrong if what you meant was that it only has vectors. It has more general objects called multivectors through pretty much the same process you get one, two, etc. forms from the wedge product.
In fact, both GA and differential forms build from the exterior algebra, and you can go from the former to the latter through geometric calculus (one key difference e.g. would be the method of reciprocal bases to compute inner products with non-orthonormal bases, rather than explicitly working out a basis and then its dual). So I'm confused about your remark regarding its alleged deficiency vs. differential forms if you pretty much reconstruct it within the GA/GC system (especially regarding working basis-free).
With regards to tensor notation in terms of calculations, if you mean all that index gymnastics, well GC still openly provides that way of computing things out from what you're used to.
What I like about geometric algebra/geometric calculus is precisely the way in which it's nothing new: it's putting everything people use in one system by clarifying the connections between these seemingly disparate systems. Even lie groups/lie algebras can be constructed rather efficiently in the algebra.
Another appealing feature of GA is its ability to make pretty transparent an old theorem from Cartan and Dieudonne that says you can view geometric transformations like rotations, and even translations (in projective geometry) as compositions of reflections.
There's other appealing features like this in terms of classifying and relating different geometries together that harken back to the Erlangen program, but my point is even in terms of concrete calculations, it's not quite right to say it's just a "more powerful version of the usual vector notation" as it includes more general objects than vectors, and still includes a lot of very similar ways of doing calculations (almost a kind of "backwards-compatibility?") you're used to with tensor index calculations, just with the added bonus of making the transition to the tensors used from vector calculus seamless, alongside other added relation to other systems made more transparent.