The main difference between
The Doctor's regeneration at the end of The Doctor Falls
The Master's death in Last of the Time Lords
The Doctor's regeneration process had already begun.
In Last of the Time Lords,
The Master is shot. The Doctor begs him to regenerate, but The Master refuses. Because his regeneration process never started, his wounds did not heal and he died.
In The Doctor Falls,
The Doctor is fatally injured, and his regeneration process begins. At this point, he can delay it (apparently by plunging his hands into snow), but he is unable to stop it.
The Doctor didn't die because he was actually regenerating, whereas The Master never started to regenerate in the first place.