"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 also used for thirds and fourths in double-stops. Our smallest finger (the 4th finger) needs to play on the lower string and if it is too flat it would cover the higher string.
Because the hand position is more sloped in sixths and octaves, we use flatter fingers. The extra stretch of flatter fingers suits octaves and tenths.
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 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....