Comparing ASME VIII Div.1 and Div.2?

Actually - ASME Div. 2 was completely rewritten for 2007 and the basis for failure of theory was changed from the maximum shear stress theory to the maximum distortion energy theory (von Mises criteria).

The most basic answer for the differences between Div 1 and Div 2 is that Div 1 is more based on design by rule (cookie cutter formulas) and div 2 is design by analysis (more rigorous calculations involved). And Div 2 "generally" doesn't apply to "high pressure" vessels either (note that it doesn't explicitly give an upper limit) but Div 3 is generally used for high pressure ~10,000 psi.

There are also numerous differences regarding material testing, NDE requirements, experimental stress analysis, user specification required, professional engineer required, allowable stresses, etc.

In short: Div 2 provides a better engineered vessel with calculated stresses closer to real stresses, combined with more rigorous testing, allows for savings in material costs (thinner parts may be used).

I would suggest digging more into this topic and possibly buying some papers from ASME to get a better idea.