A bee in the bonnet is usually situational or related to a single topic; a chip on the shoulder is more the sense of a person’s general attitude toward the world. A person with a bee in their bonnet about, I don’t know, banning trans-fats, might be perfectly reasonable and pleasant to talk with about other topics, but someone with a chip on their shoulder is probably going to be combative no matter what the topic is.
To me – yes. Bee is an external topic that bothers a person, and the person can be normal in all other aspects. Chip is some internal “insult” the world had committed against the said person, and it’s pervasive to their whole personality and world view. You can live with someone having a bee (depending on the size and viciousness of the insect), but chip carriers are totally impossible even in the shortest communication, in my experience.