The difference is that, while it is used (especially in Québec), "Je suis supposé faire quelque chose" is a a calque of the English phrase. Orally, there is not really any difference between the two, but for proper written French, you should use "Je suis censé faire quelque chose" or "Je dois faire quelque chose".

Être supposé de

On peut remplacer ce calque provenant de to be supposed to par être censé, devoir ou faire débuter la phrase par normalement, en principe, il est convenu que (suivi de l’indicatif ou de l’impersonnel), etc. : Il est convenu que je partirai ce soir (au lieu de Je suis supposé…). J’étais censée/je devais me présenter à l’entrevue hier. En principe, j’ai une entrevue demain.

TERMIUM Plus

As a complement to this other answer, where we’re presented with the influence of “to be supposed to” on “être supposé de“; in the DHLF we also have:

♦ Le passif être supposé s’emploie avec l’infinitif, pour « être censé
», par influence de l’anglais. Être supposé de (et infinitif) existe
en français du Québec.

[ Dictionnaire historique de la langue française, sous dir. A. Rey,
ed. Le Robert – supposer ]

If I understand correctly, this influence and meaning is not restricted to Québec insofar as we’re talking about the form with no “de“. There is a subtle difference between “je suis supposé faire quelque chose” and “je suis supposé de faire quelque chose“, the latter not unlike the rare construction “censé de + infinitive” (see “rem. a)” here). But you do not allude to this form.

Whereas Termium discusses the calque and offers some replacement solutions covering both the normative as well as the expectation sense, the same can’t be said of the BDL/GDT. They have an entry for supposément (which has a similar sort of influence from supposedly)¹, but have left out “être supposé“. They discuss “censé” only being constructed with “être” and followed with an infinitive verb, and meaning basically estimated/expected (see TLFi). The surprising Collins seemingly introduces a variation in the translation based on intensity i.e. required/expected to.

What will happen of this influence is uncertain but having “supposément” around in Québec surely makes the construction more usual there; different strokes for different folks. Something with the passive voice is cumbersome with either of censé/supposé and the preferred active voice with a verb such as “devoir“, which is semantically more straightforward, trumps them both when writing in my opinion. But like both Termium and the BDL explain, there is value in accuracy, as well as in variety.

_{1. It should be noted that “supposément” (1971, DHLF/Rey) has been in use for more than 50 years in Québec, despite having been sharply criticized initially; it now appears in the TLFi. Another example where opinions differ was that of the green thumb: Termium, GDT, TLFi).}