Your question pertains to ASME Section IX, so my comments will be restricted to that code section. Bear in mind that Section IX is not a standalone document. The construction code, i.e., Section I, Section III, Section VIII, etc, may modify the requirements of Section IX.
As noted by others, essential variables have an effect on the mechanical properties of the weld. A change in an essential variable beyond the ranges permitted by Section IX will affect the mechanical properties, so the procedure has to be requalified.
The nonessential variable does not have a profound influence on the mechanical properties if the weld is sound, i.e., free of unacceptable discontinuities. The nonessential variables provide specific direction to the welder.
An example will help illustrate the point. The F number is an essential variable. If the manufacturer decides to switch to a filler metal that falls into a different F number grouping, the welding procedure has to be requalified. The filler metal classification is a nonessential variable, so if the manufacturer decides to switch from DCEP using an E6010 electrode to AC and an E6011 electrode, an editorial change to the WPS is all that is required. Changing the electrode does not materially affect the mechanical properties in the example provided. However, the change in the electrical current, i.e., DC to AC, requires a change in the electrode classification. That is information that is useful to the welder to prevent the wrong electrode from being used.
A basic assumption of ASME is that the individual assigned the task of designing a system or in this case writing a WPS is assumed to have the training and experience needed to fulfill the function. As you may have noted, that is not always the case when dealing with issue relating to welding. ASME's code sections are not "how too" manuals. There are certain requirements that must be met; certain restrictions that must be observed, but the code section does not tell the reader how to design a system or how to write a WPS. Scant direction is provided on how to weld. The selection of the proper filler metal is left to the manufacturer. The manufacturer determines proper joint designs with appropriate tolerances. The ranges of the nonessential variables are left to the discretion of the manufacturer. Again, little, if any direction is provided by the various code sections relative to how to do anything. It is incumbent on the manufacturer to ensure the individual assigned to developing welding procedures understand the technology as well as the code requirements.
Many of us have reviewed WPSs that are supposed to meet ASME Section IX, but fail in the practical aspect of being a viable WPS. The result is that welders simply ignore the WPS because they have learned from past experience the WPS is useless from the welder's viewpoint.
A thorough review of the WPS provides us with some insight about the technical competence of the vendor. A well-written WPS is an indication there is someone on staff that is technically capable or the vendor has access to someone, i.e. a consultant, who is competent to provide the needed direction. The listing of reasonable ranges for the nonessential variables requires a certain level of technical expertise that is can be missing from the WPS. A poorly written WPS or a purchased Standard Welding Procedure is a reasonable indication the vendor does not have the infrastructure needed to support their welding operations.
Best regards - Al