The Master, like the Doctor is the same person whatever face he wears (or in the case of Pratt/Beevers doesn't wear). He is essentially the universal sociopath, disdainful of everyone other than himself,throwing extra disgust towards inept authority figures (usually by shrinking them to death so they match the mental image he has of them). He wants to control things because he can't see anyone else as equipped for the job as himself.
Delgado portrays this in an urbane manner as does Derek Jacobi.
He will politely enquire as to the state of Miss Grant's Coccyx while planning to send a time devouring Atlantean god to kill her and the Doctor.
The Master was originally devised to be a Valeyard type off-shoot of the Doctor's own mind. Both Delgado and Pertwee play their roles as bon vivant gentlemen. One is the hero, albeit the sort of hero who would rather not stick around and the other is the sort of villain who comes up with the idea of killer plastic daffodils and is more than willing to put it to use. Both are snappy dressers, where the Doctor has an eye on the fashions of the past the Master seems to prefer then futuristic Nehru jacket look.
The Master got his TCE around the same time that the Doctor's sonic screwdriver became a multi-tool rather than just a device for removing/ replacing screws.
Ainley wanted to play the Master as dark and twisted but he was constantly over-ruled and forced to camp it up, Simm suffered practically the same fate where as Roberts seemed to revel in camp.
Only the decayed form played by Pratt and Beevers is lacking in some form of twisted charm.
There is some debate as to whether originally the characters of The Monk and The War Chief were meant to be earlier versions of the Master.
In Terror of the Autons the Doctor dismisses the threat posed by the Master by calling him "an unimaginative plodder" which matches the Monk
and the War Chief certainly looks the part.
Indeed the War Chief (also urbane and polite and trying to conquer the universe for it's own good) wears a pendant that was previously seen in The Dalek Masterplan by Zephon
who looks more than a little like the Pratt/Beevers Master
.
Like the Monk and the Delgado Master works with the "stupid tin boxes"
which would seemingly execute him in the TVM.
Yet the First Doctor (in The Five Doctors) claims to have no memory of him even though they were supposedly both at the academy together.
The books and audios however do not follow this theory and have the three villains clearly separate people.
Interestingly in The Armageddon Factor we meet two other characters who could have easily have been the Master but for some reason were not.
Drax is another Prydonian contemporary of the Doctor. A criminal who sells weapons for personal gain and is skilled at Matter Condensation (only in this case not killing the shrunken person). He works with the Doctor against the agent of the Black Guardian called the Shadow who looks like this
when the Master was still looking like the above.
Finally you have Salyavin the boyhood hero of the Doctor who was a notorious renegade apparently locked in the Time Lord prison of Shada because of his dangerous powers of hypnotism and his ability to move his will into another being (something the Master does too).
He escaped by causing his guards to forget he existed (a ploy also used by the Master).
So you have a pantheon of pseudo-Masters.
Even the Rani (her name meaning Queen) is sort a female version of the Master. More amoral than evil. She mocks the Master for his pathetic feud with the Doctor but past incarnations of the Doctor mock each other when they bump into each other in much the same way.