As I understand it, the Pancharatnam–Berry phase first arises in the adiabatic approximation for the evolution of a quantum state. For the evolution of a quantum state parameterized by the set of parameters $\lambda_i, i = 1, \dots, N$, if the state evolves slowly enough (the details of when the adiabatic approximation holds I don't want to get into), then the projection of our state onto the eigenbasis of our Hamiltonian holds as the state evolves, plus a phase factor.
That is to say, given some initial state $$|\Psi(0)\rangle = \sum_k c_k|\phi_k(0)\rangle, $$ evolving according to the time-dependent Hamiltonian $H(\lambda(T))$,the state at a later time (when the approximation holds) will be $$ |\Psi(t)\rangle = \sum_k c_k e^{i \gamma_k(t)}e^{i \theta_k(t)}|\phi(\lambda(t))\rangle. $$ Here $\theta_k(t) = i/\hbar\int_0^t E_n(\lambda(t'))dt'$ is the dynamic phase and $\gamma_k(t) =i \int_0^t \langle \phi_k(\lambda(t')) | \dot{\phi}_k(\lambda(t')) \rangle dt'$ is our geometric phase.
If the change of our system with time is periodic, usch that after some time $T$ we get $\lambda(T) = \lambda(0)$, then the geometric phase $\gamma_k(T) = \gamma_B$ becomes the Pancharatnam–Berry phase. It's roughly analogous to parallel transporting a vector along a sphere. If I'm not mistaken, one of the important features of the berry phase is its gauge invariance.
Here is where my question comes in: Why is this important? Why does this specific phase coming out of periodic evolutions hold such importance. How might this be more important than just the general feature of a geometric phase in adiabatic evolutions? Or what important applications and uses of the Berry phase have come about?