Ability to quickly/easily cash large checks from people writing checks on their accounts at another bank, and/or to avoid fees or annoying security measures at another bank in similar circumstances.
Access to branches or no-fee ATMs, particularly in different towns or states where one bank doesn’t have branches or allied no-fee ATMs.
Access to different services of a bank that are done better by one bank than another. For example, their might be a bank that you don’t prefer for some reasons, but has a service that other banks don’t have, or don’t do as well (for example, keep various foreign currencies on hand, or offer free cashiers’ checks, or whatever).
Take advantage of offers such as @YARNLADY mentioned.
As @jeruba mentioned, to deal with temporarily blocked accounts due to whatever.