'stick' is implicitly threatening, and carries with it a secondary metaphor which, since it is subordinated, is all the more ominous

You don't need a stick to dangle a carrot, but a stick will make your point stick a little bit better.

Any donkey who has been around the block a few times has seen the same stick dangling the carrot, dangling over his ass.

If he's a smart ass, he will always choose the carrot.

It's like serving a head of lettuce on a platter. It won't improve your salad, but it might improve your manners.