What are the most significant differences between Plato and Aristotle on the notion of "how should human life be lived"?
Plato, Gorgias, 482-4, 488-500 and Aristotle, Nicomachean Ethics, Book I 1094a- 1103a
One significant difference is that Plato separates pleasure from the good. He argues that both the coward can feel just as much pleasure from his cowardly acts as the hero can from his heroic deeds. Socrates therefore argues that the good is separate and distinct from pleasure.
Aristotle, on the other hand defines the greatest good as happiness and sees that pleasure is a part of happiness. However, it's not the biggest part of happiness. Just like Plato, Aristotle sees the ultimate good as a life lived well, and since happiness is the ultimate good, happiness is more of a condition of the soul rather than just the mere experience of pleasure. It is our virtuous actions that lead to happiness, which is the ultimate good, or a life lived well.
In addition, Aristotle chooses not to define virtues, but instead chooses to address his argument to an audience who has been "brought up in good habits"(Bk. 1, Part 4). Therefore, the audience would already know what basic virtues are, such as justice, courage, and generosity ("Aristotle: Traditional Virtues and the Skeptic").
Plato, on the other hand, does chose to define certain virtues as examples of "living well." He argues that wisdom, health, justice, and self-control define a life lived well.
I believe that there is an important and fundamental methodological distinction that divides the views of Plato and Aristotle. Aristotle begins his N. Ethics by saying that this is a practical inquiry and as such does not, can not, derive demonstrative or conclusive results. Rather, as a practical guide ethical inquiry is an ongoing, open ended discourse regarding how to live a good life. Plato is penned as offering more definitive conclusions that rest on his understanding of Justice. According to Plato's Theory of Forms - which Aristotle criticises here and elsewhere - the philosopher-King knows what Justice is and erects the Republic as a close approximation to it (bear in mind that like all that resides in the world of appearance, the implementation of Justice would meet with some resistance, sometimes intentional and other times accidental).