Vantaggi
Harvard faculty are the movers and shakers of the academic world; Harvard staff are smart and well-intentioned people who are hard-working, detail-oriented and precise. It's rare to find your co-worker shirking his/her duty, or doing shoddy work. It's all about the brand. You can't beat Harvard's brand. Harvard=excellence=hard and careful work.
Svantaggi
If you are not on the faculty, you are not integral to the core mission of the university (research and education). You have to be content to be low person on the academic totem pole: Harvard, like most universities, is hierarchical. A major downside is that the attitude toward inter and intra workgroup communication varies widely from school to school and from department to department. In some offices at Harvard, you can go for years without being introduced to others in your 25 person department who are outside your 5 person division. This every-tub-on-its-own-bottom philosophy can lead to process wheel-reinventing; unless a faculty member champions a particular innovation, no one else will learn of it. Harvard Medical School's finance, grant management, education, facilities, and human resource offices all have good reason to share information internally via newsletters and updated websites, yet to my knowledge none of these groups reaches out to share information with faculty and staff on a regular basis (probably because they have to spend so much time arguing procedures/policies with their counterparts on the Cambridge campus). Also, it is very difficult to find a better job within Harvard unless you are staying in the same school - inter-school transfers/promotions are very rare.