Design by Rule (Part 4) provides rules on how to design a pressure vessel to Section VIII, Division 2 using closed-form equations and other strict rules. Protection against the most common failure modes are assumed to be achieved when the rules are followed - fatigue generally excepted.
Design by Analysis (Part 5) provides rules on using numerical analysis (typically finite element analysis) to design a pressure vessel to Section VIII, Division 2. There are rules to explicitly demonstrate Protection Against five failure modes: Plastic Collapse, Local Failure, Collapse From Buckling, Cyclic Service: Ratcheting, and Cyclic Service: Fatigue.
Design By Analysis requires specialized knowledge and experience that takes years of practice to build, working with sophisticated computational/numerical models. Design By Rule can generally be performed using much simpler computer programs, but nevertheless requires knowledge and expertise.