Converting maximization to minimization in aproximation algorithms
One connection is that maximizing some function $f$ is the same as minimizing its reciprocal $1/f$. You can use this to convert between maximization and minimization.

The general problem of maximizing a function $f$ under some constraints can be very different from the corresponding minimization problem (of the same function $f$). A case in point is MIN-CUT versus MAX-CUT. Whereas MIN-CUT is in P, MAX-CUT is NP-hard to approximate better than some constant.