What are some simple examples I can use to demonstrate the power of geometric algebra over "everyday" vector algebra?
An alternative way of thinking of this question might be: what example demonstrates how geometric algebra simplifies a tedious problem in vector algebra?
One big advantage is in the conception of some geometric transformations.
So-called "rejections" are a good example. For instance, given a vector $v$, the part of a vector $a$ that is orthogonal to $v$ is $a - (v \cdot a) v^{-1}$. This defines a linear map. When you get to finding the rejection on a blade $V$, you might, in ordinary vector algebra, have to find an orthogonal basis for the blade, and then subtract out components one at a time. If $v_1, v_2, \ldots$ are such an orthogonal basis, then it looks like this:
$$a - (v_1 \cdot a) v_1^{-1} - (v_2 \cdot a) v_2^{-1} - \ldots$$
Geometric algebra offers the more compact $(a \wedge V) \cdot V^{-1}$ instead.
Rotations are, in some sense, too well studied for GA to offer a massive advantage. Nevertheless, I believe the rotor viewpoint of rotations makes it easier to make the leap from Euclidean geometry to, for example, Minkowskian geometry. The rotor viewpoint of rotations quickly allows one to derive the equations for Lorentz boosts in special relativity.
Vector and vector calculus identities are often easy to prove with the help of grade projection.
For instance, for scalar field $\psi$ and vector $A$, consider the identity
$$\nabla \times (\psi A) = (\nabla \psi) \times A + \psi (\nabla \times A)$$
To prove it, you might have to resort to writing the cross product in terms of the Levi-Civita tensor and do some index manipulation. The GC way of doing things just uses grade projection:
$$\nabla \wedge (\psi A) = \langle \nabla (\psi A) \rangle_2 = \langle (\nabla \psi) A \rangle_2 + \langle \psi (\nabla A) \rangle_2 = (\nabla \psi) \wedge A + \psi \nabla \wedge A$$
GA/GC offers a different perspective on linear maps. Thinking of the determinant of a linear map $\underline T$ as the action of that map on a pseudoscalar is almost mind-blowing. Finding traces as the divergence of a linear map with respect to its linear argument is similarly weird when you're used to just summing a diagonal. These concepts help convince the student that traces and determinants (and other invariants that become exposed through differentiation) are real, meaningful quantities and entirely independent of basis.
GC's perspective on integration sheds some light on differential forms, also. I've run into several people on this very site who seem to think that the differentials in an integral are the same as basis 1-forms. GC shows emphatically this is not the case, as what the differential in an integral contributes is a tangent $k$-vector: e.g. $dV$ being a tangent 3-vector in 3d space.
GC makes it easy to talk about monogenic functions--functions that obey $\nabla A = 0$ for any arbitrary $k$-blade-field $A$. Conventional vector calculus fixates on harmonic vector fields instead, because the conditions $\nabla \cdot A = 0$ and $\nabla \times A = 0$ can't be married together.
GC also makes it easy to talk about Green's functions for $\nabla$ as a result. For instance, using the Green's function for $\nabla$, we can write, for a vector field $F$,
$$F(r) = i^{-1} \left[ \oint_{\partial M} G(r-r') \, dS' \, F(r') + \int_M G(r-r') \, dV' \nabla' F(r') \right]$$
That surface integral term would be horrendous to describe in vector algebra.
