What are the biggest changes when transitioning from graduate student to professor?

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...

I would say the biggest change is the necessity of multi-tasking. As a Ph.D. student or even post-doc, your responsibilities are focused and limited. You're working on a very narrow research question. Your teaching load, if any, is light. However, when you are a professor, you have numerous responsibilities with considerable weights.

1. Teaching: Your department might require you to develop and teach a new course. It could be time-consuming to develop a new teaching plan as well as notes. In addition, you might have to teach multiple courses per semester (depending on the discipline, usually between two and three courses). This makes the exam and assignment grading as well as holding office hours a fixed part of your schedule.
2. Research: Contrary to your Ph.D. dissertation, as a professor, you're usually involved with multiple projects. For example, you might advise or co-advise multiple students with different thesis topics. This makes keeping up with every one of them (e.g., advising, revising report/papers, helping with experiments, etc.) a time-consuming task. Also, the responsibility of managing research projects with industry and/or academic partners is of very importance.
3. Finding and securing funding and research grants: When you are a student, somebody else is securing the necessary funding to pay for your salary. When you are a professor, you are that person. This could be time-consuming and hard, especially if you have no prior experience with the necessary processes.
4. Service: Service includes many tasks such as reviewing journal/conference papers, organizing events (e.g., seminars, workshops, and conferences), examining theses/dissertations, etc.
5. Administration: A huge amount of your time, as a professor, will be spent in an endless series of meetings (such as group, department and faculty meeting). In addition, taking an administrative position would be a huge game changer, which would definitely affect your teaching and research plans.