Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$.
The associated Q-matrix is defined as
$$q(x,y) = \frac{d}{d t}p_t(x,y)\vert_{t=0}$$
and offers an infinitesimal description of the chain.
What are advantages and disadvantages of investigating the Markov with either the Q- or the P-matrix?
Are the two descriptions any different in the case of $S$ being finite or do the methods only differ for $S$ being countably infinite?
Those are two equivalent descriptions of the same semi-group. You mentioned how to deduce $q$ from $(p_t)$, note that, conversely, $p_t=\mathrm e^{tq}$ for every $t\geqslant0$.