The difference between transient and steady state probabilities is only that, transient probabilities are time dependent and steady state probabilities are not?
(1) A transient probability distribution is the distribution $\mu_t$ of the system at a given time $t$. (2) A steady state probability distribution $\pi$ is defined by the fact that if the distribution of the system is $\pi$ at a given time, it is still $\pi$ at any later time. (3) In general, $\mu_t$ does depend on $t$, the only case when it does not, being when $\mu_0$ is stationary. (4) Even when $\mu_0$ is not stationary, it often happens that there exists a unique stationary distribution $\pi$, that this $\pi$ is relatively easy to compute, significantly more than $\mu_t$, and that this is fortunate because $\mu_t\to\pi$ when $t\to+\infty$, for every $\mu_0$.