I don’t think you can translate this sentence without a context.
In some situation your proposition would be right, even if I wouldn’t use it without emphazising it like “Différentes personnes sont, de facto, différentes” ; in some it just doesn’t seems right and would be better translated by “Les gens sont différents” or “Personne n’est pareil” (be careful there, this is not the same “personne” as in “personnes”, this is the “personne” meaning “no one”).
All of these have more or less the same meaning but sounds better depending on the context.

For the meaning difference between “gens” and “personnes”, in my opinion it doesn’t apply here because of the “différent” which implies that you take individuals and not the group itself.

Personne is appropriate but your suggestion is unnatural. If it means “every different people is different from each other”, I would translate the English sentence by:

Chaque personne est différente.

If it means “People that are different (from the norm) are indeed different”, I would translate it by:

Les personnes différentes… le sont !