Sections I, III Classes 2 and 3, VIII-1, and XII all use Tables 1A and 1B, because those design codes use the Maximum Allowable Stress - S - in all of their calculations.
Section III Classes 1, Tc, and SC all use Tables 2A and 2B, because those design codes use Design Stress Intensity - Sm - in all of their calculations.
Section VIII-2 uses Tables 5A and 5B because that code uses Maximum Allowable Stress - Sm - in all of their calculations.
The above three allowable stresses are all based on different theories of failure and different design margins. API-579/ASME FFS-1 has a good summary of some of these differences in Table B1.4.
Which value (and Table) you use depends on which Code sends you to Section II. The values are not interchangeable.
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