These two words sound the same and both come from roots meaning “not separate,” but they have very different meanings.
Indiscreet means “tactless.”
Indiscrete means “not divisible into separate pieces.”
Indiscreet refers to a manner of behavior; only people and their actions can be discreet or in-discreet (literally, “not discreet”). Discreet means “acting in such a way that people will not notice”:
If you must answer your phone during the preformance, please be discrete and go outside.
So, indiscreet describes an action that should be discreet (not drawing attention) but which is not discreet. In other words, a public performance is not ‘indiscreet’ because it is not expected to be discreet, but even a very small action can be indiscreet, depending on the circumstances. Here's indiscreet used in a sentence:
It was indiscreet for him to smile at his mistress in front of his wife.
While indiscreet means “not-separate” in the sense of “in front of other people,” indiscrete means “not-separate” in the sense of “not having separate parts” or “not appearing to have identifiable parts”:
It is difficult to hear separate words in a foreign language, which may sound like an indiscrete stream of sound.
It may help to remember that discrete means “separable” as in “able to be distinguished”—such as different numbers.