Sensitivity-Block sensitivity conjecture - Implications
Here is what Scott Aaronson has to say on the subject:


  What makes this interesting is that block-sensitivity is known to be polynomially related to a huge number of other interesting complexity measures: the decision-tree complexity of $f$, the certificate complexity of $f$, the randomized query complexity of $f$, the quantum query complexity of $f$, the degree of $f$ as a real polynomial, you name it.  So if, as is conjectured, sensitivity and block-sensitivity are polynomially related, then sensitivity—arguably the most basic of all Boolean function complexity measures—ceases to be an outlier and joins a large and happy flock.


Checking other relevant literature doesn't offer any other compelling implications:


Nisan and Szegedy describe the question but offer no motivation at all.
Kenyon and Kutin mention that is is a "natural open question".
Gotsman and Linial give a somewhat contrived equivalent problem (Conjecture 5.33 on page 18 of the following paper).
P. Hatami, Kulkarni and Pankratov, in their comprehensive survey on the problem, also offer no motivation, but they have several equivalent formulations. For example, the sensitivity conjecture is equivalent to the conjecture that the parity decision complexity of a function is polynomially bounded by the sensitivity. Conjecture 5.31 on page 17, due to Shi, is one reformulation that doesn't mention sensitivity at all.
Ambainis, Bavarian, Gao, Mao, Sun and Zao state that the conjecture "originates from the theory of complexity measures of Boolean functions and
decision tree complexity", and generally offer the same type of motivation as Scott Aaronson does. Their recent preprint is the last word on the conjecture (as of December 2014).