What is the difference between "sum" and "existo" verbs? Would "Cogito, ergo sum" be equivalent to "Cogito, ergo existō"?
All the classical Latin dictionaries (Lewis and Short etc etc) I have consulted treat existo and exsisto as alternative forms, with two main meanings. The first is to step or come out, emerge, appear, often but not always used with the nuance of originating or arising. The second - an extension of the first meaning - is to be visible: to exist with the nuance of being discernible. Most of the citations given for the use of the word in this second sense are from Cicero and the implication for me is that "existo" had a narrower (and perhaps more learned) range of use.
Sum would be, I think, the more normal way of translating I am; existo would appear to emphasise coming into being or being manifest.