Unless you have reason to believe the school's services have a serious flaw in the process, I'd use them. The fact that there is a service connecting your school and the company suggests that someone in the career services part of your school has already done some legwork connecting the company and your school. Going through this connection may streamline the process somewhat and at the very least I doubt that it will hurt.
The only reason I would go the other way is if you had reason to believe there was some fault in the service or process offered by your school. For example, if you have seen the ball get dropped in other cases, were company systems didn't get the application, it came through all messed up, or the student didn't get any response for the company because of some flaw. Then I'd use the company offerings, because the case of error in transmission is too high.
Having worked both ends of this (been a college grad using the college system, and being a hiring manager being fed the college submissions) - I can say that in most cases the reason your college is asking for less details than the company website is because they can automatically supply much of this detail - your name, your address, your GPA, your program, and probably much of the "profile" is the equivalent to what the company asks for from the general public application. There may be a few outliers (for example a company may ask "can you get/do you have a security clearance?" when your school does not) which is likely because they know that the baseline for a college grad is a certain answer (pre-existing security clearance is typically a "no"). If you think you have such a specialty case where you have a value add that the college system does not offer a field for it, then state it in your cover letter.
Why not both?
I'd recommend against the "try both!" approach. They are both going to the same place, so I would only try the secondary approach if you thought the first had failed. If you hit both at once, you may go to 1 person, but you may end up caught in the corporate vortex where your resume gets processed separately by two different people, eventually getting merged only at some later point where someone goes "WHAT???" in which case, you don't necessarily stand out in a good way.
The company offers this position in multiple locations simply because they don't know where the best candidate for the job will come from. They opened a special channel to your college as a courtesy because they have some past history with your school. Chances are, they expect that candidates from your school will be a good fit, or at worse as good as anyone that applies from the general public. They opened the public position because it may just yeild the best candidate.
SUMMARY: When in doubt, try to come from the most elite population, using the least noisy channel.