Bernoulli explicitly models the presence/absence of a feature, whereas Multinomial doesn't.
Note that a naive Bayes classifier with a Bernoulli event model is not the same as a multinomial NB classifier with frequency counts truncated to one.
Bernoulli models the presence/absence of a feature. Multinomial models the number of counts of a feature. Here's a concise explanation.
Whereas the binomial distribution generalises the Bernoulli distribution across the number of trials, the multinoulli distribution generalises it across the number of outcomes, that is, rolling a dice instead of tossing a coin.
What does this imply for us? Multinomial NB cares about counts for multiple features that do occur, whereas Bernoulli NB cares about counts for a single feature that do occur and counts for the same feature that do not occur.
This means that, for example, Multinomial NB will classify a document based on the counts it finds of multiple keywords; whereas Bernoulli NB can only focus on a single keyword, but will also count how many times that keyword does not occur in the document.
So they do model slightly different things. If you have discrete multiple features to worry about, you have to use Multinomial NB. But if you only have a single feature to worry about, then you can make a modelling choice based on the above.
Not the answer you're looking for? Browse other questions tagged naive-bayes-classifier or ask your own question.
How to use TFIDF vectors with multinomial naive bayes?
When to use Multinomial Naive Bayes?