Anscombe's concern in Modern Moral Philosophy is that without a divine law theory the idea of "moral obligation" would no longer be what one might term today a "thick concept" if by that one means what Brent G. Kyle calls a "substantively descriptive" concept: (page 15 of the linked file)
Now let us remember that "morally wrong" is the term which is the heir of the notion of "illicit," or "what there is an obligation not to do"; which belongs in a divine law theory or ethics. Here it really does add something to the description "unjust" to say there is an obligation not to do it; for what obliges is the divine law - as rules oblige in a game. So if the divine law obliges not to commit injustice by forbidding injustice, it really does add something to the description "unjust" to say there is an obligation not to do it.
We generally have the same normative concepts, thick or thin, such as "untruthful", "unchaste" or "unjust", as did Aristotle. But unlike us, Aristotle did not have the concept of moral obligation that adds an obligation not to be "unjust". Anscombe references Aristotle not because his vocabularies are different but rather to provide an example: "...you can do ethics without it [moral obligation], as is shown by the example of Aristotle." (page 7)
Anscombe claimed the law conception of ethics, that is, moral obligation, is what makes us, when we fail, "bad qua man (and not merely, say, qua craftsman or logician)". To have a law conception of ethics one would need to "believe in God as a law-giver; like Jews, Stoics, and Christians". (page 5)
However, today one tries to retain "a law conception without a divine legislator" and the divine law becomes transformed into conscience, norms, pleasure, laws of nature, or contracts (pages 11-12). However, these lead to a justification of injustice that she illustrates with the "judicial punishment of the innocent" under some circumstances. To get around this would require a philosophy of psychology, replacing the divine law and legislator, but that is "conspicuously lacking". (page 1)
To address the question:
But I still do not understand why she thinks her vocabularies are superior in making moral judgements.
The only difference between the Aristotelian vocabularies and those today is the idea of moral obligation inherited from a divine law conception of ethics. Aristotle did not have this. We have it today as an empty concept that we assume exists nonetheless. Otherwise the vocabularies are the same. We could continue an ethics without moral obligation the way Aristotle did.
She does have a "complaint" which might make her position superior if she is correct (page 16).
Our desire to retain moral obligation without keeping a divine legislator forces us to come up with some alternate justification for that moral obligation, such as, conscience, pleasure, social norms, consequences, intentions or physical laws. However, these lead to a potential justification of injustice under some circumstances, that is, to justifications where "one 'ought' perhaps to commit an injustice" under some circumstances.
Because of this potential for justifying injustice found in modern moral philosophy, she recommends ceasing to pursue the idea of moral obligation and follow an ethics based on Aristotle's example.
Anscombe, G. E. M. (1958). Modern moral philosophy. Philosophy, 33(124), 1-19. https://www.pitt.edu/~mthompso/readings/mmp.pdf
Kyle, B. G, "Thick Concepts" Internet Encyclopedia of Philosophy https://www.iep.utm.edu/thick-co/