Plato's and Aristotle's views on politics were very different. Aristotle rejected many of the building blocks of Plato's politics: the theory of Forms (in the Metaphysics), the universal idea of the Good (in the Ethics), the value of Communes (in the Politics).
Further, unlike Plato, Aristotle distinguished "theoretical" philosophy (physics, metaphysics) from "practical" philosophy (ethics, politics). Aristotle agreed with Plato that theoretical philosophy was part of the highest (and potentially happiest) form of human life. But unlike Plato, Aristotle held that theoretical philosophy had no practical value. He even saw its impracticality as a sign of its value: theoretical philosophy has no goal outside of itself. Accordingly, theoretical philosophy played almost no part in Aristotle's ethics and in his politics.
It is plain then that Science [= theoretical philosophy] is the union of Knowledge and Intuition, and has for its objects those things which are most precious in their nature. Accordingly, Anexagoras, Thales, and men of that stamp [= theoretical philosophers], people call Scientific, but not Practically Wise because they see them ignorant of what concerns themselves; and they say that what they know is quite out of the common run certainly, and wonderful, and hard, and very fine no doubt, but still useless because they do not seek after what is good for them as men. (Ethics Book VI)
Aristotle argued against regimes ruled by a small (minority) class, on the ground that such a regime cannot remain stable for long. Aristotle advocated a broad ruling class (the middle class) and a wide participation of citizens in the management of the state.
One factor that Aristotle preserved, as it were, from Plato's politics was the importance of public education. The corriculum of Aristotle's public education system included some forms of gymnastics and of music. Aristotle also suggeted that future rulers and legislators will study ethics and political theory, similar to his own. But he did not relate studies like higher mathematics or theoretical philosophy to the management of the state, as did Plato.
Another political aspect which seems to have been common to Plato and to Aristotle, was that they both viewed the desired state as outwardly peaceful, utilizing military force only for defense, and avoiding the occupation of other states.