Ockham concludes that essence and existence speak of the same thing, but the first as a noun and the second as a verb, that is, they are no more distinct in reality than, say, “John is taking a lunch” and “John is lunching”, where we describe one and the same thing first by a noun and then by a verb.
Ockham sees a sufficient likeness between res, essentia, and entitas to move freely from one to the other, and a sufficent likeness between esse, existere, and existentia to do the same. He takes a real distinction as being between a one thing and another.
Here are his arguments from Summa totius logicae 3.2.27. I’m paraphrasing them and not translating them ad litteram.
1.) If existence is distinct from some thing, it is either substance or accident. All sides agree it is not a substance (one substance can’t be the existence of another) but if it is an accident it must be one of the nine genera. Both are absurd.
2.) If existence is unified to a thing, it either makes something intrinsically one or not. If not, the thing would be one only accidentally; if so, then essence is matter and existence is form. Both are absurd.
3.) If essence and existence were two things, God could preserve one without the other. But both are impossible. Ockham does not say why, but presumably he means it is impossible to preserve the existence of a thing without either the existence of the thing or the thing that exists.
4.) Note that if X does not exist, it cannot be anything at all – a fortiori it cannot be an essence. Therefore an essence that does not exist is not an essence. So if you say “essence may exist or not exist, therefore they are distinct” it is no different than saying “essence may or may not be an essence, therefore it is not an essence”.
5.) When the holy authorities say that God is ipsum esse, all that needs to be meant is that God exists necessarily and not from something else while the creature is contingent and from something else.