Semantics aside, I’m assuming that your question is essentially “why do we use a matrix formulation of quantum mechanics rather than a continuous variable/differential equation/integral formulation” (I may be wrong and would welcome clarification) and nothing to do with the interaction picture and the like, which some other answers seem to be touching upon.
Largely, we use the bra-ket notation of Dirac, so we might write quantities such as $\langle\psi|\phi\rangle$. These are very abstract entities. You can choose to represent these abstract entities using either a vector/matrix formalism of an integral formalism. It doesn't matter.
However, we typically choose to use vectors just because it tends to make the maths easier. Vectors have, somewhat more obviously, properties such as orthogonality built into them, so I don't end up evaluating integrals all the time. Particularly when I want to talk about a two-level system, I could describe it using two continuous functions $\psi_0(x)$ and $\psi_1(x)$, but it's just overkill.
The sort of problem that I studied as an undergraduate would talk, for example, about the infinite square well. You'd be given some initial state $f(x)$, and asked what it looks like at some later time. Perhaps you have to evaluate the probability that the particle is on the right-hand side of the well. By far the quickest way of doing this is to decompose
$$
f(x)=\sum_n\alpha_n\psi_n(x),
$$
where $\psi_n(x)$ are the stationary states of energy $E_n$. The only thing I really care about is the discrete set of values $\alpha_n$, which are most conveniently stored as a vector. Then the solution would be
$$
\sum_n\alpha_ne^{-iE_nt}\psi_n(x).
$$
Next, I decompose the state of being on the right-hand side of the well as
$$
\sum_n\beta_n\psi_n(x).
$$
The probability that the final state is on that side is
$$
\left|\sum_n\beta_n^\star\alpha_ne^{-iE_nt}\right|^2.
$$
The only integrals that I need to do are to evaluate the $\alpha_n$ and $\beta_n$. The rest of the calculation is more easily done using orthogonality/linearity properties which are all present in the integral form, but are harder to see. So you very quickly start abstracting away from the integral form and are naturally led towards linear algebra.