Probably the primary reason that public school districts and universities (as well as other governmental entities) utilize separate foundations is that they are likely to provide greater freedom, flexibility and control for the program since they can usually operate outside the normal governmental processes.  The foundations can hold permanent endowment funds that may be difficult or impossible to maintain within the government.  They may attract grants from private foundations that won’t fund governmental agencies, but will make a grant to the school foundation.  They may avoid union labor contracts, and union provisions for construction projects.  They can usually avoid government bidding and procurement regulations.  They may be able to accept anonymous gifts that might not be possible with a governmental agency subject to a public records act.  (See Nonprofit Issues®,  Lessons from Litigation, August 16, 2008.) They are somewhat insulated from current political controversies and less likely to have their mission overturned by a single election.  The foundation board can determine the programs it wants to initiate or support without requiring the political consensus that would be necessary in a governmental entity. Donations can be restricted and protected from other demands of the school district.

I am sure there are other reasons, but these are the ones I hear most frequently. There are those who would say that some of these “advantages” are significantly anti-democratic, but they are sufficiently real that new governmentally-related foundations are being formed on a regular basis.