Why is isospin so useful? I'm currently reading about isospin in nuclear physics, and I know how to calculate it, and all the math, but I'm actually not sure WHY it is so useful?
Can anyone come with some examples where not using isospin quantum numbers and such would be terrible, or is that not even the case?
 A: Isospin is useful because the strong interaction which is important in the creation of nuclei is charge blind. The strong interaction sees a nucleon  and does not see its charge. The effect of the charges enters as higher order corrections to how a nucleus is formed. As a first order approximation treating the proton and the neutron as two faces of a nucleon simplifies calculations . See for example the binding energy per nucleon.
A: Isospin is useful approximation when dealing with groups or families of particles with nearly the same mass.
For example, the proton and neutron are practically degenerate (have nearly the same mass, why we call them nucleons) but differ by charge. If you ignore the mass difference you can say that each nucleon has isospin 1/2, and the difference between the two nucleons is you assign one with isospin along the z axis of +1/2 and the other -1/2, very analogous to spin angular momentum. 
The same goes for the pion triplet, you assign them an isospin of 1, and then assign each a projection along the z axis (1, 0, -1).
A: From a practical point of view, when performing nuclear structure calculations one often stores matrix elements of the nuclear interaction in the form $\left\langle a b | V | c d \right\rangle$. Where $a,b,c,d$ label different orbitals. If we distinguish protons and neutrons, then we need to store separate matrix elements for
$\left\langle pp | V | pp \right\rangle$,
$\left\langle pn | V | pn \right\rangle$,
$\left\langle pn | V | np \right\rangle$, and
$\left\langle nn | V | nn \right\rangle$.
This can be rewritten in isospin formalism by coupling the two nucleons to $T=1$, with $T_z=1,0,-1$ and $T=0$ with $T_z=0$. So we trade four proton-neutron matrix elements for four equivalent isospin matrix elements. However, if we make the approximation that the nuclear interaction depends on $T$ but not on $T_z$, then we only have to deal with two matrix elements rather than four.
The gains are even more substantial if one wants to include three-body forces.
The isospin formalism also makes it clear that there should be isobaric analog states, that is, states in a chain of isobars (nuclides with the same mass number A) which have very similar structure and energies -- after correcting for Coulomb effects -- except that they differ by the ratio of protons to neutrons. In the isospin formalism, these states correspond to different projections $T_z$ of the same state.
