OK. When I say 'Stiffness' I'm referring to the classical Hookean definition of stiffness, that being the k in Hooke's Law: force (or stress) = stiffness * deflection. Stiffness is also known as flexiblity.
In our case here, stiffness in a pipe stress model is a representation of how flexible the compressor flanges are. Not just the compressor flanges, but the piping, pipe supports, compressor foundation, etc.
If you recall the basic beam flexure formula S=My/I, it can be rearranged to solve for a stiffness k which is deflection per unit load. Another version of Hooke's Law is S=E*epsilon, where epsilon = strain, aka deflection.
Lawyers... let's say there's a failure, with subsequent litigation. The other side will bring in SMEs that will ask you if the stress model of the piping system was properly configured, to include a representation of the flexibility (stiffness) of all components of the system: piping, flanges, supports, foundation, everything. If the piping system model is not up to snuff, you will be hung out to dry because the case will be made that you did not exercise what is called the 'reasonable man standard' of care and custody, i.e. what would another similarly-qualified person have done in this situation.
We have to remember these systems are covered by the ASME B31.3 piping code. That code has very specific requirements for piping flexibility analysis. Some type of flexibility analysis is mandatory by the code. Horse sense is a perfectly acceptable way to do pipe stress. I do it all the time. But I make damn sure that it is applied to the proper situation. Rotating equipment is not typically one in which I would apply horse sense because there is no way with horse sense to quantify how much load is being applied to the flanges, and without a properly-configured stress model I can't conclusively show whether the system complies with the allowables, especially on gas compressors because they always have hot discharge piping between the discharge and the aftercooler (if an aftercooler is present).
If you, as the person of record on the project, can say that you have installed X number of nearly identical systems with zero failures, then that method will satisfy the intent of the code.
Bottom line: Whether or not you apply stiffness/flexibility in your stress model, or whether or not you need to even run a stress model, is a risk analysis and code compliance issue.