Excellent question. I dare anyone to provide an answer based on real life side-by-side experimentation. I myself do not have this answer. I use them pretty much interchangeably based on whatever I happen to have on hand or whatever is on sale.
I actually did this when I really started getting into all-grain recipes - I compared red vs. white wheat malt & flaked wheat vs. torrified wheat. By all means trust your own nose/palate, but here are the notes I took during my side-to-side comparisons:
Red Wheat Malt (Briess): Wonderful flavor w/candy-like sweetness and very smooth mouthfeel; subtle wheat/malt aromas
White Wheat Malt (American): Candy-like sweetness w/light honey flavors; very chalky mouthfeel; faint honey & caramel aromas
Flaked Wheat: Thick, creamy mouthfeel w/subtle sweetness; faint "wheaties" aromas; zero fluid release - begging for a stuck sparge
Torrified Wheat: Thick, creamy mouthfeel w/subtle sweetness; faint "wheaties" aromas; good fluid release
So flaked vs torrified wheat flavors and aromas were no different (to me), but I always use torrified just to avoid the possibility of a stuck sparge, and to receive more runnings.
I also tend to use Red Wheat Malt instead of the White, simply because I didn't like the chalky mouthfeel of White Wheat Malt (but it definitely has more of a honey character to it).