I have seen people use both fari and farti for the verb "to do", so I'm asking whether they are the same, or whether they have different meanings.
When you say that farti is being used for to do, my guess is that you are referring to a translation of the phrase “how do you do?” as “kiel vi fartas?”. In this case farti is only a translation of to do in the very specific sense of being in either a good or bad state. I think a closer translation in English of farti could be to fare, as in “how do you fare?” except that the Esperanto word doesn’t sound old-fashioned.
The word fari is much more common and is a better translation of the more general sense of to do (as well as to make). Also note that fari is transitive as in the general sense of to do, ie you do something (fari ion) whereas in the other sense you are just either doing well or doing badly (fartas bone, fartas malbone) with no object.