As I understand it, the original motivation was to study CREW PRAM (consecutive read exclusive write parallel RAM) model. In this model, several processors compute a function with shared memory access, but with no write conflicts. Stephen Cook and Cynthia Dwork and Rudiger Reischuk ("A lower time-bound for parallel random access machines without simultaneous
writes.") showed that $\Omega( \log(s(f)) )$ is a lower bound on the number of steps required to compute $f$ on CREW PRAM. Let $CREW(f)$ denote the minimum number of steps required to compute $f$ in CREW PRRAM. Thus, Cook, Dwork, and Reischuk showed that
$CREW(f) = \Omega(\log s(f))$
Later, Noam Nisan tweaked the definition of sensitivity to characterize $CREW(f)$ up to constant multiplicative factors. The tweaked definition is, of course, block sensitivity. I.e.,
$CREW(f) = \Theta(\log bs(f))$
The natural question then arises whether $CREW(f) = O(\log s(f))$? This is essentially the sensitivity conjecture.
Whether nowadays the implication of sensitivity giving a tight characterization of $CREW(f)$ is compelling or not - I cannot judge. As pointed out in another response, it turned out that block sensitivity is included in a large class of polynomially related complexity measures, while sensitivity is not known to belong to that class.