There are two types of academic positions at many institutions, a tenure and clinical track.
For both, benefits of an academic position include having residents do as much of the clinical work you want them to do, not having call, and (usually) not having to worry as much about a referral base/competition/etc unless you want to get involved in those aspects of the department. For the tenure track, in exchange you publish and (ideally) write grants that get funded or participate heavily in cooperative group trials. Lots of writing.
For the clinical track, in many departments you'll be expected to carry more of the clinical load while still spending some time teaching, possibly publishing a bit with residents, etc.
In a private practice you'll be taking care of patients and doing all the work yourself. If you have a nurse practicioner you'll have someone to take some of the busy work off your hands, but generally practices that are busy enough to warrant an NP mean the radiation oncologist is hustling.
Starting salaries are usually pretty similar between most academic institutions and private practices that follow a partnership-type structure, though will be higher for hospital-based private practices, urorads groups, and HMO organizations (Kaiser). Post-partnership income will generally be higher in private practice, though some academic physicians and those in hybrid practices do very well.