Professional fee coding is the billing for the physicians. The facility coding is billing for the facility and the equipment (and things like room charges when pt is admitted).
In your example, the physician who read the xray would bill for the xray with the -26 modifier to indicate professional services. The hospital would bill for the xray with the -TC modifier attached to indicate technical services.
This situation is assuming the physician is employed by someone other than the hospital though. He's on staff at the hospital but not paid by the hospital. He does his billing under his own group NPI/tax ID number. The hospital owns the equipment, therefore the physician bills for the prof component only.