what is the relationship between entropy and variance?
They measure different things.  Given the variance, you can solve for $p$ and compute the entropy, and vice versa, but they have different interpretations and different applications.  You can learn a bit about the meaning and applications of these notions in the corresponding Wikipedia articles: https://en.wikipedia.org/wiki/Entropy_(information_theory), https://en.wikipedia.org/wiki/Variance.
Analogy: if we have a sphere, I could measure its radius, or I could measure its volume.  They both reflect the size of the sphere, so you could as, what is the advantage of radius, or of volume?  Answer: they measure different things, so depending on what you want to do, one or the other might be more useful.