A cortado is a double shot cut with about the same amount of warm milk and almost no foam, around 120 ml in all. A flat white has roughly twice as much milk to the coffee and a thin layer of silky foam.
The differences
Ratio: a cortado is about 1:1 coffee to milk; a flat white is about 1:2 or more.
Foam: a cortado has next to none; a flat white has a thin, glossy layer for latte art.
Origin: the cortado is Spanish ("cut"); the flat white is Australian and New Zealand.