What's really important in this discussion is understanding the (potential) failure modes, and how the different stress levels play into them.
As had been said, the lower values are typically used for deformation-sensitive components (flanges, etc). However, you need to have an understand of
why that is.
From Appendix 1:
Two sets of allowable stress values are provided in Tables 1A and 1B for austenitic materials and specific nonferrous alloys. The higher alternative allowable stresses are identified by a footnote to the tables. These stresses exceed two‐thirds but do not exceed 90% of the minimum yield strength at temperature. The higher stress values should be used only where slightly higher deformation is not in itself objectionable. These higher stresses are not recommended for the design of flanges or other strain sensitive applications.
At this point, you may still be wondering why this is an issue. Well, the stress-at-a-point in a vessel component is not uniformly at or less than the allowable stress. In fact, the localized stresses can be substantially more - as much 3 times more (or even greater, in certain circumstances). So, in these types of situation, it is acknowledged that there will be higher stresses - and in the case of the "higher" allowable stresses, almost assuredly, the stresses will exceed yield.
(Of course, an appreciation of the full elastic-plastic stress-strain curve of these materials in needed, as they are usually quite ductile, yet exhibit a low proportional limit).
So, you need to understand the strain-sensitivity of your component. If some plastic deformation is acceptable/permissible, then you are very much welcome to use the "higher" allowable stress values for your design, and still be confident that your vessel will have an equivalent design margin against burst as any other vessel. And, based on experience, the "lower" allowable stress should be used if your component is strain-sensitive. In fact, you should probably do some calculations (FEA is NOT required) to demonstrate to yourself that the selected allowable stress is appropriate; sometimes for strain/deformation-sensitive equipment, even the "lower" allowable stresses are not low enough.
So, figure out, first, what the potential failure modes for your component are, and then select the allowable stress accordingly. And, of course, read the footnotes and the Appendices.