Charlemagne's Holy Roman Empire encompassed the countries of central Europe, from what we know as Poland to Austria and from France to the Russian border. In contrast, the larger Roman Empire extended from Scotland to all sides of the Mediterranean Sea.
One massive difference is the length of time that these two empires lasted for. What was key in the Charlemagne empire was the way in which the personality of Charlemagne held it all together. When he died, so did his empire. In the Roman Empire, by contrast, although of course emperors were incredibly important, at the same time, when one emperor died the empire itself continued.
Under Charlemagne, there was no real centralized state, simply a collection of counts held in sway by fear, obligation, or mutually beneficial arrangements. There was little uniformity in law, no imperial policy, and ultimately Charlemagne ruled by force of arms and personality. One similiarity might be the de facto existence of a state religion- Charlemagne compelled all conquered people to embrace Christianity.
One difference was the "international" versus local trade. During the Roman Republic and Empire, trade spread all around the Mediterranean, into the Middle East, Africa, and Europe. By Charlemagne's time, that had all but disappeared. As Western Europe went feudal, the only trade that existed, if it existed at all, was between geographically local areas.
The main difference between the Roman Empire and Charlemagne's Holy Roman Empire was that the Roman Empire was based in Rome, then Constantinople and Charlemagne’s empire was based in Central Europe, including what we now call Germany.
It was feudal. The Carolingians were not one united empire with a bureaucratic system and a standing army. Instead, they were a feudal empire with lords who had real power.
It had more of a Germanic form of law. It allowed things like wergeld as compensation for wrongfully killing someone.
It was more dependent on the Catholic Church for its legitimacy. Of course, the Roman Empire was Christian by the end, but that wasn't really what bound it together or gave it its identity.