I thing the phrase "time-independent Schrodinger equation", though widely used, can be a cause for confusion because it implies that it is in some sense a different version of the Schrodinger equation, which isn't true. The Schrodinger equation is a first-order differential equation
$$i\hbar \psi'(t) = \hat H \psi(t) \qquad (\star)$$
where $\psi(t)$, which is an element of whatever Hilbert space underlies your theory, can be loosely thought of as the state of the system at time $t$. This equation tells you how to evolve your state vector forward in time.
If $\psi(t)$ is an eigenstate of $\hat H$ with eigenvalue $E$, then this equation becomes extremely easy to solve:
$$i\hbar \psi'(t) = \hat H \psi(t) = E\psi(t) \implies \psi(t) = e^{-iEt/\hbar} \psi(0)$$
If $\psi(t)$ is a linear combination of such eigenvectors, then the solution is similarly very simple. If $\psi(t) = \sum_n c_n(t) \phi_n$ with $\hat H \phi_n = E_n \phi_n$, then
$$\psi(t) = \sum_n c_n(t) \phi_n =\sum_n e^{-iE_n t/\hbar} c_n(0) \phi_n$$
The fact that $\hat H$ is self-adjoint means that every state can be written as a linear combination of its eigenstates$^\ddagger$. Therefore, in order to understand how to evolve any arbitrary vector forward in time, we need only find the set $\{\phi_n\}$ of eigenvectors of $\hat H$ and then expand the initial vector in terms of them. From there, time evolution is simple - each eigenvector evolves via the corresponding phase factor $e^{-iE_n t/\hbar}$.
Therefore, we turn our attention to finding all of the eigenvectors and eigenvalues of $\hat H$. This requires finding all pairs eigenvector/eigenvalue pairs $\{\phi,E\}$ which are solutions to $\hat H\phi = E\phi$. Once we have all such solutions, we have all the ammunition we need to evolve state vectors forward in time. This equation $\hat H\phi = E\phi$ is what we call the time-independent Schrodinger equation, but I really dislike that name.
As you can see, the TISE really of completely different character than $(\star)$ - the former being the eigenvalue equation for the Hamiltonian operator, and the latter telling us how to evolve a general state forward in time. Solving the TISE is an important and very useful step in solving the "general" Schrodinger equation, but they aren't different versions of the same thing.
$^\ddagger$This assumes that the spectrum of $\hat H$ is discrete, and that $\hat H$ is not explicitly time-dependent. Further complications arise if either of these two assumptions fail, but the spirit of this answer remains the same.
