Do schools prefer their application to the common application if they accept it as long as you fill out the supplement and send them all the information they need?

Answer

All colleges that accept the Common Application agree to treat all applications equally. In fact, many now use only the Common App (including Harvard). The application you use is not as important as the care and thoroughness that you put into completing it. And most importantly, the strength of your transcript matters most, followed by other criteria including your essay, SAT/ACT scores, letters of recommendation, extracurriculars, etc.

Answer

They prefer you to fill out and submit their application because it shows your personal interest to that specific college/university instead of them thinking that you just filled out one application and forwarded it to various colleges.