Other than the fact that a starship has phasers and an aircraft carrier doesn't, how would you characterize the differences in their combat capabilities?
That's a book-length topic. Give me a year or so! But let's look at some of the things that aren't immediately obvious. We can break this down into two major dichotomies: organic versus inorganic capability, and known versus unknown threats.
Organic versus inorganic capability refers to whether a vessel like the Enterprise provides its own firepower and logistics, or relies on other ships. The U.S. Navy organizes its surface combatants around the carrier strike group, or CSG. A CSG notionally includes a carrier, three or four escorts, and logistics support ships, with perhaps a submarine also assigned. So the carrier's air wing can range over thousands of square miles, the escorts range out from the carrier, keeping threats away from the big ship, and all the ships in the CSG are networked together so that whatever is seen by one can be seen by all. The carrier itself carries not only the air wing, but also functions as a command-and-control node, a fueling station, a shipping depot, a complete medical and dental facility, and a host of other functions as well. In other words, a carrier strike group has a lot of organic capability.
But there're a lot of things that a CSG can't do for itself that must be performed by assets based ashore, including logistics, intelligence, communications, guidance, planning, and other services. The CSG operates in waters that have been sailed for thousands of years, using charts that are the best the U.S. government can buy. All in all, there are perhaps 10,000 people directly involved with the day-to-day operations of a carrier strike group, spread across several time zones, operating in a well-understood environment.
Contrast that situation with that faced by any of the various Starfleet captains we have seen. The various Enterprises we have seen are solo performers. In deep space, on a long-duration mission, it can't count on the support of other Starfleet ships and assets. So it must have its own organic capabilities, such as performing scientific analysis or repairing damage to the ship. I'm sure Starfleet must have other vessels somewhere that are designed to operate in groups, but we haven't really seen them. While we occasionally see Enterprise and her sisters brought together for some event, usually a battle, it's clearly a rare event.
In terms of known versus unknown threats, I can't emphasize enough how almost everything that the U.S. Navy does involves known factors such as geography, international law, and capabilities and intentions of adversaries, while everything Starfleet does involves unknowns. That's really the norm for any science fiction involving new alien threats. The U.S. Navy has the benefit of knowing that the other side flies aircraft with jet engines, and hence we can make infrared-homing missiles that will seek out those engines. And if you are really good, your missiles have all sorts of counter-countermeasure seeker logic that ignores decoys or any other heat source that that doesn't look exactly like a jet engine, which is why that fancy F-22 owns the skies against the Russians and the Chinese.
But if the aliens from Independence Day really came to Earth, our missiles may not be able to lock on to their ships, because the features that make your missile smarter at dealing with decoys make it dumber in dealing with something for which it has not been programmed. The seeker logic just assumes it's a decoy trying to spoof it, and ignores it. So, if your alien would-be target also has that other staple of science fiction, the cloaking device that absorbs radar waves, then you're pretty much left with harsh language. Yes, maybe you can get a shot off with your gun, but the guns are tied into the radar, too, so maybe not.
That's the situation faced by Starfleet. They may be operating in known space, dealing with well-understood adversaries, or they may be encountering a new race with new technological capabilities for the first time, perhaps in an area of space that Starfleet has never visited before. So, Starfleet has to have systems that can deal with new technologies and capabilities from aliens it has never encountered before. That's what strikes me when I put my naval analyst hat on: all of the systems in Starfleet are incredibly flexible. Of course, for dramatic reasons, sometimes they fail.
What about the way that each ship would fight?
A U.S. Navy carrier faces well-documented threats and generally well-understood technology. When it engages those threats, it might do so from hundreds of miles of away. Combat, if it occurs, takes place between ships that can't see each other.
For reasons best explained by the nature of TV shows and movies, the fictional Enterprise fights at engagement ranges more appropriate to sailing ships than a modern warship.
The combat I see in Star Trek really doesn't look like anything the U.S. Navy does today. What it resembles is a cross between World War I surface ship combat and modern submarine combat. It's like submarine combat because the ranges are close and the engagements are generally one ship versus one ship. It's like World War I combat because the target is in plain sight and the trajectories are flat. However, where combat in Star Trek differs from those models is the extreme precision of weapons, so that called shots targeting specific features of an enemy vessel, such as the engines, have become commonplace.