The key differences in the implementation of the algorithm / in the passing of the messages are as follows:
messages in BP are passed only once; in contrast, messages in loopy BP (LBP) are passed iteratively until "convergence is reached" (convergence means that the belief does not change anymore from one iteration to the other).
in a Bayesian ...
Cycles are redundant, that's all. And it is pretty easy to show if you think about the transitivity.
A -> B -> C -> A implies A -> B and B -> A
So the directionality is immediately lost (for every pair in the loop).
A -> B -> C -> A is also deterministic, the multiplier is one.
Say B = (1/3)A + e1 and C = (1/2)B + e2; ...