What are the biggest changes when transitioning from graduate student to professor? I often hear the adage that graduate school never actually trains you to be a professor. I would be curious to know what were some of the biggest hurdles assistant professors had to overcome when they first obtained their jobs. Were these mainly interpersonal and managerial skills? If so how did you sharpen your skills in those areas?

Are professors as overworked as graduate students? It is a common fact that graduate students frequently suffer from mental illness and depression. Is this the same for professors? If professors are even busier than graduate students, how do they avoid burning out? How is the stress different?
Your question can be answered partly by the different time sharing:

link

Other bigger changes for a professor: 

Much stronger independence and responsibility and expected to be successful, while balancing all of this correctly. With all its implications. Being an academic role model within and outside of your group, leading your team, but also leaving room for creativity and self-development to your group members. And this is sometimes still difficult for full professors, some apparently suffering under this pressure or trying to bypass it by shifting responsibility to co-workers, wrong managing of division of labour, scientific misconduct... without this correct balancing probably much less questions would be asked on academia.se?!

Though, in my opinion the most important and inobvious and maybe most argued change is that you are expected to give back to the society and public. It's not like you have won in the job lottery and can lean back more like for industry or civil servants positions and stick to your 40 hour contract. There is a reason why most professors work many more hours a week, it's a lifelong commitment to their job and its duties and privileges. And therefore it is a important life decision one should make before starting to pursue an academic career ending within professorhip.

On the other side, coming up, defining and realizing good/competitive research ideas over decades can be mentally exhausting, also a sabbatical nowadays has become an off-academia phenomenon and in academia typically meant having really time for finding and outlining new research ideas and questions apart from the other time-consuming task above in the graphic.

Soft skills are necessary and helpful like in every other job, but not sufficient. Many universities offer such courses specially designed for students and postdocs and getting such certificates is helpful (project management, academic english,...)

Professors are typically not taught to teach, they learn this themselves and in my opinion every student has experienced this result in a better or worse way. But evaluation of teaching of a applicant for proffesorship is often a crucial criterion to become full professor. Therefore, a proficient language level (if no native speaker) is often necessary and good feedback/evaluations by the students, if the professor is not outstanding in acquired funding and publication track.

This article on a "A year in the life of a new professor" on jumping from lab to classroom is also a good read to get a up to date impression of young professors

Disclaimer: I'm no professor, postdoc on the road maybe to become one...