I think money is highly signficant because of what it signals. If a school offers you money it's a signal that they think highly of you. If, by sad chance, you happen to have a bad first year, a school that offers you money is much more likely to keep you around for another year. However a school that decided to not offer you money implictly signaled that you were not among their top choices that year. So if you have a bad year, that school is likely to take the grades as confirmation that they were correct: You are a weak student and hence should be dismissed. Even if they don't dismiss you from the program, it likely will be much harder to get a decent professor to offer to serve as your disseration chair.
In the past I've recommended that a student go to a school ranked 15 that offered him a signficant stipend over going to a school ranked about 10 that did not offer him money. I felt at ease making that suggestion...but if the choice had been the 15th ranked school versus the 2nd ranked school without money,....well, that choice would be much more difficult.