In "yiwei"/"creer" case, Juan believes that they are going to promote him (but his belief is not very well calibrated and is likely false). yiwei/creerse asserts something about the truth value of the belief, in addition to what the belief is.
In the "doubts" case, Juan believes that they are not going to promote him. There is no assertion regarding the truth value of that belief.
Quite a bit, actually. It shows that Juan is aware of it, whereas in the Spanish equivalent he may actually believe it, even though it still is false. In a way you are very much illustrating the GP's point. And if I got it wrong then I am doing the same :)
TIL thanks, but the evidence is weak I'm afraid. English isn't my mother tongue, and it's 6am here. I misread that this "strong connotation" was about the subject (Juan) and not about the object (Promotion).
How does yiwei/creerse differ from "Juan doubts that they are going to promote him"?