What are the differences (if any) of the tenure-track job hunt if you're a professor compared to a PhD student I'm asking mainly for the field of computer science.

It is expected and often encouraged for PhD students to market and advertise themselves during several months it takes to apply and interview for academic positions. However, it seems that when non-tenured/tenured faculty switch institutions, it's normally done silently and even the faculty's own students might now know about the search until it is a fait accompli. 

I'm wondering how the process of switching universities might be (or is) different for someone who is already a professor. What is the procedure for professors planning to switch universities?
The difference is that when you are a PhD student, everybody knows you won't be around for long, and everything is (or should be) planned accordingly. You are assigned duties knowing that you will move on relatively soon.

Someone with a more permanent position is assumed to be around for much longer. They can take on longer time duties, and other people may start depending on them actually sticking around.

So, if a faculty moves to another place, some people are likely to get upset. Now, people upset with you across the country is not a big deal; but if you advertise it and fail to get the new position, you will have upset people in your own corridor.

Another factor is that, as soon as faculty announce they are likely to move away, people are going to start dividing up their loot: I'll take half of your lab, because you are leaving anyway; I'll teach your favourite course, because you are leaving anyway; we won't buy you the new equipment because you are leaving anyway... Again, if the deal is off and you have to stay, you have lost part of your lab, teaching, and opportunities for equipment or funding. Note that the second factor is also true for positions on soft money, that understandably, no one should count on them sticking around.

And lastly, the good old pride: "so, you think they are better than us?"