A center is like a department, except it doesn't have any resident students. The old CMT had students working with it, but they had to have some other home department (usually the CL program in Philosophy or the CS PhD program in SCS).
We still have such students, but now, as the Language Technology Institute, we are an institute, and have resident students as well.
It is unclear to me whether there is any semantic difference between an institute and a department. At the time we chose the name, I was told the only difference was stylistic.