The equations of geometric algebra can be represented using matrices (perhaps with a non-euclidean metric for the inner product depending on application), so in that sense you don't gain the ability to do anything that you couldn't do with regular linear algebra.
The "big wins" are that the notation is simpler and easier to reason about, and everything is coordinate-free. As an example, there is a model of 3 dimensional euclidean space that can be represented using a 5-dimensional GA (with 4,1 metric signature). Within this model, you have lines, planes, circles, spheres, etc; but you also have rotations, reflections, translations; and all of these things are just standard elements of the mathematics with no special cases. So you can multiply a translation and a rotation and you get back the composition of those two things. Or you can multiply a sphere by a translation and you get back a sphere that has been translated by the specified amount. Or if you want to smoothly interpolate a rigid body motion (translation+rotation) between two points, you can take the logarithm of that motion. And again, this can all be done symbolically without resorting to coordinates - except at the very end, perhaps, if you want to map it onto an actual coordinate system :)
In terms of people applying it to more "advanced" subjects like Minkowski space or differential geometry - I'm not so sure. I haven't spent too much time looking into that.
The "big wins" are that the notation is simpler and easier to reason about, and everything is coordinate-free. As an example, there is a model of 3 dimensional euclidean space that can be represented using a 5-dimensional GA (with 4,1 metric signature). Within this model, you have lines, planes, circles, spheres, etc; but you also have rotations, reflections, translations; and all of these things are just standard elements of the mathematics with no special cases. So you can multiply a translation and a rotation and you get back the composition of those two things. Or you can multiply a sphere by a translation and you get back a sphere that has been translated by the specified amount. Or if you want to smoothly interpolate a rigid body motion (translation+rotation) between two points, you can take the logarithm of that motion. And again, this can all be done symbolically without resorting to coordinates - except at the very end, perhaps, if you want to map it onto an actual coordinate system :)
In terms of people applying it to more "advanced" subjects like Minkowski space or differential geometry - I'm not so sure. I haven't spent too much time looking into that.