Visible crew/equipment: After Doctor Strange experiments with the magical doorway in the Sanctum Sanctorum, he leaves it on a desert view and we cut to a shot panning across all the artifacts. At the very start of the shot on the right hand side of the screen is a crewmember standing in the shadows looking bored. Really blatant if you turn up the brightness, but easy enough to spot if you know what you're looking at. (00:56:00)
Other mistake: When Dr. Strange rushes a patient to brain surgery they show him and Dr. Christine Palmer operating without surgical masks. When they cut away and come back they're wearing masks. Also, Dr. Palmer is an ER doctor, and her assisting him would never happen in real life.
Factual error: When Dr Strange first exits the London Sanctum there is a blue USPS letter drop box on the corner. London Postal boxes are a different design and red.
Factual error: When Tilda Swinton dies and is lying on the operating table, her pupils are not dilated as they would be immediately after death.
Continuity mistake: When Kaecilius is attached to a wall just his hand is free, but in the next shot now his head and other hand are free too.
Other mistake: After the fight where his sling ring is stolen by Kaecilius, he has a sling ring in his possession and uses it to jump to Hong Kong.
Answer: Medically, he could not heal his hands. He spent his entire fortune trying to return to being a surgeon through experimental procedures. Once he completed his training to become a sorcerer, he realised there was more to the world than he realised. The Ancient One correctly stated that he was a surgeon for his own ego; he wanted to prove he was the best and helped people for that reason only. By joining the sorcerers he was given a greater purpose and chose this over returning to his former life.
His hands were healing, just not quickly enough for his needs. This was showing his egotistical attitudes.