"Curved fingers are very good for articulation but don't stretch very far. Flatter fingers can stretch much further."
Curved fingers are good for articulation, especially in fast passages. They are used for thirds and fourths in double-stops.
Flatter fingers give a warmer sound in lyrical passages (the softer finger pad in itself gives a warmer sound and also helps make the vibrato warmer). Flatter fingers are used for sixths, and their extra stretch suits octaves and tenths.
Flatter fingers will not be as flat as shown here on the table top: just slightly less on the fingertip and more on the pad of the finger than the more curved finger shapes....