In U.S., why do many engineering departments care about professional engineer registration? I have seen many engineering departments want professional engineer registration. Why do they care?
There are 2 major theories about credentials: human capital theory and signaling theory. Under HCT, a license (such as a PE) shows that you have accumulated a credible amount of knowledge (you must graduate from an accredited engineering school) and experience (you need to have worked for 4 years after your bachelors to sit for the PE exam). Under signalling theory, the PE shows that you have done what it takes to legally call yourself an engineer.  One interesting comparison of the differences of HCT and ST is The Career Consequences of Failing versus Forgetting. You may know just as much as another person, but the one of you that passes some hurdle signals to prospective employers that the hurdle passer is the better candidate. This is because hiring a person is trying to predict future behavior/success with limited information, and many people use signals as heuristics. 

You will also find out that universities hire people who have degrees. A cynical view is that they have a vested interest in maintaining the supply of people who get degrees. A signalling theory viewpoint is that universities think degrees are important enough that they only hire teachers who have them. 

In many fields of engineering, your working career will be very short if you do not pass your PE. Civil is one such. Other engineering fields, such as Electrical (which is mine), typically have state exemptions for manufacturing, so very few EEs take their PE. When I was younger, I was quite opposed to licensure. Now, I see it as a way to distinguish myself from other candidates. One interesting blog post that inspired me to sit for my PE exam is this one. Another is a dissertation (which is not online) titled "Hiring and Inequality in Elite Professional Service Firms".

My advice is to take your EIT and PE exams as soon as practical. Some US universities require you to take your EIT exam during your senior year (as in they won't issue your diploma without passing it). 

Disclaimer: I am registered to take the PE exam this April.