Forgive me, I'm thoroughly confused by that dichotomy. How are they different? Approaching from bayes rule or a "maximum likelihood" approach produces the same results.

The problems of the filter are present in both.

Well, the derivations are different, and your comment seemed to imply that the maximum likelihood perspective was easier to understand.

The result is identical, the understanding is different. I would suggest that the Bayesian perspective leads to insights like the UKF [1] which IME is all round much better than the apparently better known EKF for approximating non linear systems.

[1] That is, it is generally easier to approximate a distribution than a non linear function.