What is the difference between quasi-static and moderate dissipation load applications in Abaqus?

First and foremost, checking the documentation should always be your first step.

Second, those three options control the amount of dissipation (numerical damping) that the implicit integrator (Hilber-Hughes-Taylor) introduces. If you ran a model of a simple pendulum by giving the pendulum a nudge, the oscillation will damp out quickly with the quasi-static option whereas oscillations will continue for longer with a transient fidelity option. When you choose those options, the code picks a couple of parameters that the developers thought were reasonable based on the problems they ran. You have the option of opening up the keyword deck and changing the parameters according to your need.

I oversimplified it in order to get the intuition across.

*********************************************************
Are you new to this forum? If so, please read these FAQs:

Check the documentation chapter "Implicit dynamic analysis using direct integration". Quasi-static is a special type of analysis used for problems that are time-dependent but slow enough for the inertial effects to be ignored.

IceBreakerSours, how would moderate dissipation affect the pendulum?