Researchers from the General Counsel Roundtable on outside counsel management practices defined these two terms in a sensible way, and this blogger will adopt the definitions.
A fixed fee is an amount a law department agrees to pay a law firm to handle all work in a given area for a period of time. For example, all SuperFund preliminary assessments in the United States during 2010.
A flat fee, by contrast, is a set fee to be paid a law firm for a particular matter. For example, $150,000 to obtain the zoning variance for a warehouse extension.
This information comes from the Met. Corp. Counsel, Vol. 18, Jan. 2010 at 18 (See my post of March 1, 2008: fixed or flat fees with 36 references.).
A unit fee, I should add to this trio, is a fee a law department agrees to pay to a law firm for each matter of a pre-defined sort. For example, each application for a tier 1 patent through the first office action will be paid $7,500.