In series 6, during The Impossible Astronaut, the age difference between the two Doctors we see (the Tesselecta and the Envelope #1 Doctor) is said to be 200 years old. In the Time of The Doctor, when the TARDIS returns, The Doctor says that it has been 300 years. My question is, during The Impossible Astronaut, he hasn't aged a day in 200 years between 2 versions of himself. And yet, here, in 300 years, he is suddenly all old and wrinkled. So how come he ages so ripe in Time of The Doctor and not in The Impossible Astronaut? Is this a discrepancy, or is there some aspect of TimeLord ageing process that I'm unaware of ?
We don't know, but we can make educated guesses.
We've seen one other Time Lord hit the end of his regeneration cycle: The Master, in Old Who. When he reached the end of his last body, the Master fell apart. Almost literally:
So when the Doctor reaches the tail end of his last lifespan and appears to suddenly age rapidly, there is precedent for it being just because he's finally actually reaching the end of his life: after aging 200 years in Impossible Astronaut, he then added another 300+ years on top of that.
Time Lords age more slowly than humans. The Doctor is not aging faster as soon as he gets to Trenzalore. He is just aging. As you said, 200 years doesn't show a wrinkle on him. But age is cumulative. He doesn't just start form zero every time that he counts his age. By "The Time of the Doctor", the eleventh incarnation is approximately 300 years old. The time that he initially spent at Christmas added another 300 years onto his age. So by that point, the eleventh incarnation would be 600 years old and finally showing age. And the time on Trenzalore after that only added more years in a very strenuous situation.
Note that we have seen one other aged Doctor, of course: the First. We only seem him old, in fact. Since that is definitely his first incarnation, we can assume that he aged normally (for a Time Lord) up to that point. Going by the other times the Doctor refers to his age, we can assume that he was about 230 years old when he first started travelling with the Tardis. This is quite a bit less than the (roughly) 600 years that Eleven was before he started showing age, though. We can perhaps speculate that, if left to age normally, each incarnation would last longer before needing to regenerate.
His regeneration energy could keep his current incarnation young but being at the end of his cycle he doesn't have the energy to sustain youth without killing himself. In the episode in which Tennant regenerated he used his excess regeneration energy to do many things i.e. grow back a hand (which also by R.E. grew into a meta-crisis), the master aged him rapidly but with the help of every human's thought he partially regenerated to regain youth; matt smith after getting a new cycle became young again even though he fully regenerates into peter later in the special. Also in Angels take Manhattan he "wasted" R.E. to heal River song's wrist which could have kept his youth. It is my guess but in the whoinverse there are many examples that support my hypothesis
I always figured it was because Time Lords don't age in the same pattern humans do. So suppose a hypothetical Time Lord regenerates into an infant. He then grows and goes through puberty, then he reaches maturity and the aging process slows down, so he appears 30 at the age of 120 (since the regeneration,) then about 35 at the true age of 500.
Then, when the Time Lord reaches a certain point, it speeds up because he's reached the end of his life.
He didn't age faster than normal, it's just that the Doctor most likely spent over a thousand years in his 11th (13th) incarnation and unsurprisingly became elderly, decrepit and weak.
This isn't hard to reconcile.
Between the age of 20 and 35 you won't seem to change much; but then just wait until you progress between 45 and 60. Oh, boy, just wait!