When we're talking about spatial rotations is quantum mechanics, why do we need to resort to $SU(2)$? Why isn't $SO(3)$ enough?
I've read that $SO(3)$ isn't simply connected, and I've read about Gimbal lock, and Double covering, but I'm falling down deeper and deeper into pure mathematics (esp. topology). Is there a "topology for physicists" explanation of what's going on here?
In classical physics you need to describe mathematically physical quantities whose meaning in related to some directions. Some of this quantities are described with vectors, their rotation can be described with the rotation matrices of $SO(3)$. However there are some quantities (like spin in quantum mechanics) which are described with mathematical objects that are not vectors but spinors. If you want to write the matrices that describe how spinors rotate in the three dimensional Euclidean space you will find the matrices of $SU(2)$.
If you are interested in why there are spinors in quantum mechanics there are many possible explanations. The wave function is represented as a certain mathematical object that has certain property of transformation under rotation. It's possible to establish a connection between the generators of the rotation of the object and the intrinsic angular momentum. It turns out that with spin $s=1/2$ the mathematical objects to describe the wave functions are two-component spinors.
It's possible to demonstrate that there are two different matrices of $SU(2)$ for every matrices of $SO(3)$ and this means $SU(2)$ is a double cover of $SO(3)$. Topologically speaking, you can find differentiable manifolds that are isomorphic to $SU(2)$ (that roughly speaking means that they mathematically are the same) and differentiable manifolds that are isomorphic to $SO(3)$. A differentiable manifold isomorphic to $SU(2)$ is the three sphere $S^3$ while for SO(3) is half $S^3$ and this can help in visualize the meaning of double cover.
In quantum mechanics, we consider eigenvalues and eigenstates of angular momentum operators, and the eigenstates are labelled $|l,m>$. $l$ is the total angular momentum and $m$ is the z-component. In fact for the purposes of my argument, consider the three angular momentum operators (Lx, Ly, Lz) restricted to a subspace of their spectrum where the "Casimir Operator" $L^2 = L_x^2+L_y^2+L_z^2$ is a constant $l(l+1)$. $L_i$ will still obey the usual commutator, $[L_i, L_j] = i\epsilon_{ijk}L_k$, but since we restricted to this subspace, they will be $d\times d$ matrices, where $d$ is the dimension of the restricted Hilbert space.
You will remember that for $l$ integer, $d=2l+1$. That is just because $m=-l\ldots0\ldots +l$. And because angular momentum is the Noether charge associated to rotational symmetry, the quantized observables restricted to this $2l+1$ dimensional subspace obey the usual commutation relation described above, we can say that those operators form a $(2l+1)$-dimensional representation of the rotation group (technically the Lie Algebra $\mathfrak{so}(3)$, and the vector space that they act on (since group representations act on a vector space) is spanned by the eigenstates $|l,m>$ for $l$ held constant. These representations are obviously all \textit{odd-dimensional}.
You will also remember that experiments suggested we consider $l$ to be half-integer, notably $l=1/2$ for electron/nucleon spins. To describe the spectrum and operator algebra of spin-angular momentum operators, we can try to copy the above argument and seek $(2l+1)$-dimensional representations of the Lie Algebra of the rotation group $SO(3)$. But since $l$ is a half-integer, this is an even-dimensional representation.
As it turns out, this is impossible. There exist no even-dimensional representations of $\mathfrak{so}(3)$. However the double cover $SU(2)$, with Lie Algebra denoted $\mathfrak{su}(2)$, does have even dimensional representations with the same commutation relation as the $\mathfrak{so}(3)$ algebra. So, we have no choice but to use those to describe the quantum mechanics of half-integer spin systems.
This has important consequences stemming from how even-dimensional "spinor" representations of $SU(2)$ behave. Rotating your spinor by $2\pi$ (which leaves space unchanged) actually causes a phase change in the spinor wavefunction, and this is a real, experimentally verified effect that stems from the fact that spinors transform under the double cover $SU(2)$ as opposed to the usual rotation group $SO(3)$.
TL;DR: We need even dimensional representations of the rotation group to describe spin-1/2 systems, and we can only get these by working with the double cover $SU(2)$.
