To partially answer your question, the copper kings of Butte ran Montana, economically and politically, for a century. Since the end of mining, Butte has been searching for a reason to exist. I am not sure they are even close to finding an answer at this point.
A quick way to compare Missoula, Bozeman, and Butte is to look up the size of the student bodies, breadth of the degree programs, size and age of the facilities, and extracurricular activities available at the University of Montana, MSU, and Montana Tech (formerly the Montana School of Mines). Most students like the same outdoor places and activities you do!