I sometimes see or hear these verbs used interchangeably.

Problem:
The verbs “flaunt” and “flout” are not synonyms.

Explanation:
The verb “flaunt” — spelled F-L-A-U-N-T — dates back to the 1560s.

When used with an object, “flaunt” means to display ostentatiously.

When used without an object, “flaunt” means to parade oneself conspicuously.

The verb “flaunt” apparently comes from the Norwegian dialectical word “flanta”, which means to show off.

I first learned the verb “flaunt” from a Braniff Airlines TV ad, in which Braniff had stewardesses proclaim “If you’ve got it, flaunt it!” to brag about Braniff’s superiority over other airlines.

The verb “flout” — spelled F-L-O-U-T — dates back to the late 1300s.

When used with an object, “flout” means to scorn, scoff at, or mock.

When used without an object, “flout” means to show contempt or disdain.

The verb “flout” comes from the Dutch word “fluiten”, which means to play a flute. That word also means to jeer, which apparently was often accomplished hundreds of years ago by someone playing a simple flute or whistle around another person.

Solution:
To remember that “flout” is different than “flaunt”, notice that “flout” looks somewhat like “flute” to remember that “flout” means to mock or to show disdain, as if you were playing a flute in a mocking way around someone else.