Under what circumstances and why is terrific broth preferred to LB broth for E.coli growth?
I have used terrific broth to grow up transformed E. coli so they grow faster.  Often people will conduct a transformation, screen the cells, and grow them up in broth for plasmid extraction.  This growth would be for 12-16 hours.  Instead a person could use terrific broth to grow up the cells and it would take may 6-8 hours for adequate cell density for plasmid extraction.  If you are trying to express a protein then I would go with LB.

Also, I learned recently that terrific broth is also preferred over LB when needing to express proteins after transformation.