What's the relationship between drag divergence Mach number and maximum operating limit Mach number?
They are unrelated definitions. Mmo is set by the designer with input from actual flight testing, Mmo is a maximum safe speed and is only concerned with practical operation and structural integrity. Mach divergence is a physics phenomena. Mmo can be much lower or higher than the Mach divergence from one aircraft to the next. Mmo has a similar purpose to Vne but Mmo is a more useful measure for high performance aircraft. 
It just happens that most commercial and large business jets are designed for high subsonic cruise so the design goal is to push divergence to higher speeds for more efficiency and Mmo ends up very close to Mach divergence.