Here is a definition from Wikipedia that is consistent with what I know (and I’ve studied them both breifly). This quote explains both systems. I included the last sentence because Rand’s admiration for Aristotle is potentially significant in talking about her ethics, although of course she did differ with him on many points.
Virtue ethics is currently one of three major approaches in normative ethics. It may, initially, be identified as the one that emphasizes the virtues, or moral character, in contrast to the approach which emphasizes duties or rules (deontology) or that which emphasizes the consequences of actions (consequentialism). The origins of this theory date at least back to Plato and Aristotle (it, arguably, has roots in Chinese philosophy that are even more ancient).
So, to sum up, virtue ethics advocates living virtuously, while consequentialism advocates making choises that produce the best outcome. Utilitarians are consequentialists, but I’d say rational egoists are as well. The basis of Rand’s ethics is that it’s in one’s self-interest to act morally, because it has the best consequences.
So it should be clear why someone would advocate consequentialism, as Rand did, to those of us on this forum. But why did Aristotle advocate virtue ethics? From Wikipedia and also consistent with what I know:
To achieve eudaimonia (happiness) one must live by what can be considered virtues such as justice, wisdom, courage, prudence and so forth. A virtue ethicist would argue that this is what all humans would rationally choose to live by. To help us achieve eudaimonia we must practice to be virtuous.
Notice the part in bold. If we assume what Aristotle meant was A virtue ethicist would argue that this is what all humans would rationally choose to live by _ because it produces the best outcome for the human _, his ethics would be both in line with consequentialism, and in line with Rand.
I like use this analogy: “virtue ethics” is to “rational ethics” as “sense of life” is to “rational integration of the world.” Virtue ethics is a shortcut to help people apply ethical principles - for example, to act justly, wisely, etc. - without having to go through detailed ethical arguments as Rand presents in her books every time they want to make a good decision.
Note: The Wikipedia quotes are, as far as I know, objectively true. But my argument that virtue ethics can be considered a subset of consequentialism, and my analogy relating virtue ethics to rational ethics as advocated by Rand, are my own arguments. As a footnote, I think virtue ethics could also be considered a subset of deontology, depending on how you back up your argument for virtue ethics.
Also, I realize Rand didn’t advocate hedonism - that happiness was the highest goal - which is what Aristotle seemed to advocate, at least based on what I know (haven’t read much original Aristotle). The commonality is that they both advocated, from what I know about Aristotle, practicing a system of ethics that produced the best result for the individual practitioner.
*Edited to add second footnote