The First Way is clearly borrowed from the argument of Physics VIII. There are significant differences between the two, but not in what they take as true, like:
1.) One ends a physics, the other starts a theology. Aristotle is trying to solve physical problems like “how are some things always in motion, and others not?” and “what is required if it is impossible for motion to cease?” and “Can movers and moved things combine form an actual infinite?” Rephrased as questions about energy, the questions are still outstanding. St. Thomas is trying to establish the minimal possible fact about deus (God or a god) in a way that can be fruitfully developed by further discourse.
Notice that if we separate the proof from either context, it will seem like an objection to say “but this isn’t real physics!” or “but this doesn’t prove (some relatively much more developed view of) God!” The first claim is false when said of Aristotle and irrelevant when said of STA; the second completely misses the point of both proofs since the first proof was meant to solve a set of physical problems and the second is targeting the minimal possible criteria for a deus.
2.) A merely unmoved mover is not adequate for Aristotle. The nature of the questions he is asking require him to argue for an unmoved mover that (a.) possesses infinite power and therefore (b.) cannot be a magnitude. St. Thomas on the other hand thinks he hits a minimum threshold of divinity when he finds an immobile reality, presumably because he is persuaded by the claim that whatever is mobile is natural, and so an immobile being is supernatural. Here again we see the difference between Aristotle trying to answer problems in physics and STA looking for a minimally-basic theological claim.
3.) The necessity and infinite duration of the universe. While STA is fine with necessary creatures, he claims A.’s arguments for the infinity of the universe beg the question. The historical account of this is known to all – STA thinks Genesis commits him to the historical-scientific claim of a finite universe, and he finds a universe of infinite time possible even if rationally implausible. His reasons for allowing this possibility are relevant to the difference between the two proofs, since STA thinks that proving that the universe comes from a god means that it arose from some wisdom that we cannot penetrate with sufficient precision to know whether its creating has always accompanied its being. In other words, proving that the universe traces back to an unmoved mover makes some physical questions impossible to answer unless the answers are revealed.