Near as I can tell, the quarter system is far superior from the student perspective. I'm talking about using 4 equal class terms instead of 2 long and 2 short terms. I don't recall the exact lengths, but a full school year (minus summer) in the semester systems is something like two 15-week classes and in a quarter system three 10-week classes or thereabouts.

Here are the differences off the top of my head:
- quarter system GPA is more meaningful because you have more samples, more data; semester system GPA can be swung by outliers
- semester system compresses the study time and is more intense due to fewer breaks with longer classes, reducing academic performance
- a quarter system is much easier for course administration and selection because they are all equal in length
- more chances in a quarter system to get those classes you need to graduate, just have to wait until the next quarter, more opportunities
- flexibility and specialization with electives is easier in a quarter system with more course offerings and more open schedule
- flexibility in starting summer work/internships/coops in either April or June/July, instead of only one option (you can sort of choose the length of your summer)
- better mission-related program entry/exit point flexibility in a quarter system
- quarter systems actually provide a spring break
- being stuck in a class with a bad professor is much more manageable if it's shorter, in a quarter system

I've probably missed a few too, but I'm not really able to think of any advantages to a semester system other than students being on campus for more of the fall/winter sporting events.