Blush, flush are comparable as verbs primarily when meaning to turn or grow red in the face and as nouns when meaning a turning or growing red in the face.
Blush implies a sudden heightening of color usually as the result of shame or embarrassment.
Blush is often extended to matters (as the dawn, the sunset, or a fresh growth of pink in the garden) that suggest a delicate glow or warmth of color.
Flush basically suggests a flooding (as by a rush of water) or a sudden brilliant flashing or glowing. In reference to the face both of these ideas are involved, for flush may imply a reddening of the face that brightens or floods the countenance sometimes transiently like a flame or it may imply the more or less persistent redness of the face that is associated with robust health, anger, elation, fever, or hard drinking.
Flush, therefore, connotes modesty or shame much less often than does blush. In extended usage flush refers to lights or colors that flash and dart and glow.