Aristotle was St. Thomas Aquinas's main intellectual influence, but in what way did St. Thomas disagree with Aristotle somewhat in his view of the “just state”?

The difference between Aristotle and Thomas Aquinas, in regard to the "just state," may be understood from their different understandings of natural law. In the case of Aristotle, natural law is tied to the fact that humans are social animals, and so the end goal is to promote the "perfect community" in which people live with others. The "perfect community" to be promoted, for Aristotle, is life in the polis (city-state) where men are free and equal. Civil law, in this "perfect community," is just insofar as it is in harmony with the law of nature (i.e., promotes virtuous existence). Thus, civil law should reference the law of nature: to lead men into virtue.
Thomas Aquinas, being a Catholic theologian, viewed natural law in reference to God’s eternal law (Aquinas's aim was to develop a Christian view of natural law). This law is derived from scripture but is also imprinted in the minds of humans and discerned through reason. The "just state," therefore, should not contradict God’s eternal law and should seek to derive and apply its precepts in ruling. According to Aquinas,

Consequently every human law has just so much of the nature of law, as it is derived from the law of nature. But if in any point it deflects from the law of nature, it is no longer a law but a perversion of law.