I sometimes hear people mispronounce one of these words as if it were equivalent to the other word.

Problem:
These two nouns are not synonyms.

Explanation:
The noun “anecdote” means a short account of an incident as an unpublished narrative.

The noun “antidote” means a medicine for counteracting a poison.

Those who confuse these two nouns probably are distracted by the fact that both nouns have “dote” — D-O-T-E — in them, the fact that both nouns start with A-N, and the fact that both nouns are eight letters long.

The noun “antidote” is constructed from the prefix “anti-“, which means against, and “dote”, whose origin essentially means given. In other words, the roots of “antidote” when assembled mean given against.

The noun “anecdote” is constructed from the prefix “an-“, which means not, and “ecdote”. The “ecdote” part of “anecdote” is a combination of “ec”, which means out, and “dote”, whose origin, as just noted, means given.

So the “ecdote” part of “anecdote” essentially means published. In other words, the roots of “anecdote” when assembled mean not published.

Solution:
Use “anecdote” when referring to a spoken story; use “antidote” when referring to a remedy for poison.