This answer is good if by "conservation of probability", you mean dynamical conversation but let me try to parse the phrase "unitary aspect of representations was important physically" in another way.
Schrodinger's wave mechanics and Heisenberg's matrix mechanics both satisfy $[Q,P] = i \hbar$ but do so in totally different Hilbert spaces. You may ask, who is right? -- or, how many other representations of this relation are there? Well, the Stone-von Neumann theorem tells you that all representations of $[Q,P] = i\hbar$ are unitarily equivalent. That is, there exists a unitary $U$ taking one representation to the other: $[U Q U^\dagger, UPU^\dagger] = i\hbar$.
So the Schrodinger and Heisenberg representation are unitarily equivalent -- but you still want to know which one is the right one. Well, it turns out that they both are. In any representation, the physical predictions are manifest as probabilistic predictions for the outcome of an event $E$ (represented as a positive operator) given the state of the system is $|\psi\rangle$, viz.
$$\Pr(E|\psi) = \langle\psi|E|\psi\rangle.$$
Now consider the predictions of unitarily equivalent representation:
$$\Pr(UEU^\dagger|U\psi) = \langle\psi|U^\dagger UEU^\dagger U|\psi\rangle = \langle\psi|E|\psi\rangle.$$
So unitarily equivalent representations are physically indistinguishable.