After reading the two cases you get a good sense of Elizabeth Parker's ability to integrate and gain status through Case A. Although Elizabeth had more authority in Case B it was clear in case A that she was building up her power. Her networking techniques both while seeking the job and once she had the position were an important part of her ability to build and wield power in the organization.
In Part 3 of the "Political Lens" we read about Henry Mintzberg's 3 sources of power with the first consisting of resources, technical skills, or specialized bodies of knowledge, the second being formal power, and the third having access to others who have power. In Case A Elizabeth selects a central project involving her in the whole department essentially giving her access in building her future power potential. Also she creates a new researcher role which offers her resources and knowledge while not appearing to take someone else's position. Elizabeth finds herself with 2 of 3 sources of power in case A and she did this all while lacking formal power.
The comparison between case A and B is difficult because B focuses on avoiding failure whereas A's completion would result in great returns for the department and more power for Elizabeth. Also we don't really know what Elizabeth Parker does at the end of case B so it is unclear of how she utilizes her formal power to avoid potential project failure.