I will provide a generic answer.
The reason they want to remain separate is for financial and administrative reasons, in whatever order you prefer. For many medical school clinical departments there is a "tax" and/or high overhead to support less profitable specialties and other institutional activities. As a "private" clinic, professional fees basically stays with the practice, which basically means with the physicians.
In addition, "medical schools" have lots of rules. This is particularly true for clinical faculty. (I will add for completeness that particularly for state medical schools, there is often a byzantine web of corporations that the physician may actually work for and at least receive a significant amount of salary from.) This can range from the number of vacation days allowed to what CME funds can be used for. A "private" clinic allows the physicians to tailor these things to their particular specialty. For example, neurosurgeons and radiologists view vacation in radically different ways.
So why do medical schools allow these affiliations? Because surgeons, as opposed to internists and EM physicians, are in relatively short supply. These specialties also add to the luster of the medical school and the medical center. They are separate because they can be. Most other specialties simply don't have that leverage. Prospective medical students are generally more impressed with "wow, what incredible cutting edge things these surgeons are doing!" compared with "wow, the tenth diabetic foot check this morning!"
As to the "day to day" life, I will leave that to an actual surgeon who works for one if they answer.
