From a "New York Times" blog post about the use of the word "foundation"
versus the use of the word "charity":
Some charities, however, have the word "Foundation" in their official names. Examples of these are the Yele Haiti Foundation, the New York Foundation for the Arts, the William J. Clinton Foundation and the Wikimedia Foundation. Despite their names, all of them are charities; they rely on donations from others to sustain themselves and the programs and services they offer. On second reference, any one of them should be referred to as a "charity," not a "foundation."
Source: http://topics.blogs.nytimes.com/2011/04/26/phrases-gone-astray-2/
It appears that nobody appears to actually follow this rule (including the "New York Times"), but I find the nuance interesting. I imagine one would perform better than the other during fundraising; perhaps there's hard data on that.
MZMcBride
There isn't any "rule" more a suggested guideline with respect to "On second reference".
Foundation is not a legal term; a charitable exemption could be granted to either a trust or a corporation. If it is "founded" a corporation might be funded by a trust established by the founder.
However; there a sense in which Jimmy Wales founded and funded the Wikimedia Foundation, but not with vast funds.
Fred