Hey there, I think I was the one who gave you the only-partly-accurate information. SOPHAS is a new thing (possibly just this year?), so I wasn't aware of it. It sounds like a great idea - it was a pain to provide essentially duplicate information to a number of schools - but my experience with online applications in general suggests that caution is good: make a test print of anything you plan to submit to make sure it looks how you want it to look, keep copies of everything, and check in with the school if you're not sure your application is complete. I think the overall cost of applying will probably work out the same, on average.
Keep in mind that not all schools of public health currently accept the SOPHAS application. There is a map showing participating schools in yellow and non-participating schools in white:
http://www.sophas.org/SchoolMaps.htm
Many schools do seem to be participating (though some of the participating schools may also have "supplemental" applications in addition to the basic information).
Good luck, and hope the SOPHAS system serves you well and makes your life easier!