The adjectives discrete and discreet are homophones which share the same Latin origin: discretus, meaning separate. They are pronounced the same way, but have different meanings.

Discrete has stayed close in meaning to its Latin origins and means individually separate and distinct:

• We can no longer view extreme incidents such as flood, drought and high temperatures as discrete happenings, but must study them in the context of climate change as a whole.
• The golf club has three discrete membership categories.
• The mechanical device consists of several discrete parts which all work together to perform a function.

Discreet has moved away from its Latin roots and means careful, circumspect, reserved, cautious and modest:

• The tables in the restaurant were well-spaced to allow for discreet conversation.
• The work of MI5 is carried out in a discreet and confidential manner.
• She implored her friend to be discreet and not to gossip about her troubles

Discretion is the noun corresponding to discreet. To act with discretion is to act in a cautious, reserved, or modest manner. In certain cases, the word is extended to mean freedom to act on one’s own judgement.

Discreteness is the noun corresponding to discrete. It is the state or quality of being discrete, separated or distinct.