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Perturbation theory expands a quantum problem around a solvable Hamiltonian. Time-independent perturbation theory expands stationary energies and eigenstates; time-dependent perturbation theory expands dynamical amplitudes and transitions.

Expanding a state in unperturbed energy eigenstates with their free phases converts the time-dependent Schrodinger equation into coupled equations for slowly varying amplitudes. To first order, the perturbation on the right-hand side acts on the initial unperturbed state.

This is the time-dependent branch of perturbation theory; its corrections describe transition amplitudes.

Time-independent perturbation theory expands the eigenvalues and eigenstates of $H_0+V$ in powers of a small perturbation $V$.

This is the stationary branch of perturbation theory.