‘When someone achieves greatness in any field -such as the arts, science, politics, or business- that person’s achievements are more important than any of his or her personal faults.’

Whether is more important professional achievements or personal faults is a very debatable topic given that both are part of everybody’s life. However, from my point of view is more important professional achievements.
The first thing we should analyze is what is more important, personal or professional life. My personal experience tells me that is more important the second one because your personal life is, in many times, conditioned by professional life. For example, if you are successful in your job you would be more confident in terms of what you do with your personal life. Moreover, professional success makes you feel better with yourself and this is transmitted to your personal life too.
Additionally, if a scientist finds the cure for AIDS he or she will be remembered for this achievement and nobody will remember him because of his failure in his or her marriage or any other personal problems. Therefore you will be remembered for your professional achievements given that these ones a less commonly achieved by normal people. It is what makes you special and different from the rest of the people.
Finally, with professional achievements you will get the recognition of the society and colleagues. And with personal failures only are known by your closest friends and your family. Then professional achievements are more widespread and are known by more people.
In summary, achieving professional greatness is more important than your personal fails because people is remembered by society and history by that thing that you made and nobody else could do it.