Space carriers that launch fighters don't appear much in Star Trek, except for a few episodes of Deep Space Nine. Carriers and fighters are more of a fixture in Star Wars and Babylon 5. So the carrier Enterprise and the starship Enterprise are designed according to two very different concepts of warfare?
Very much so. In the core Star Trek universe -- that is, what we have seen on TV and in the theaters rather than the novels and games -- fighters don't really make sense. That's because in the core universe, bigger is generally faster, while deflector shields make it difficult or impossible for small ships to have ship-killing weaponry. In the real world, fighters work because they are machines of the air, where they can have tremendous performance, and because they carry a punch disproportionate to their size. Neither of these preconditions exist in Star Trek. Again, the closest model is World War I naval combat. This hasn't stopped some of the ancillary material, like the various war games set in the Star Trek universe, from adding spacecraft carriers that use warp shuttles as fighters.
How would life be different -- or the same -- on a starship versus an aircraft carrier?
There are a lot of similarities, and a lot of differences. Both U.S. Navy and Starfleet ships have dedicated, highly disciplined crews of skilled professionals. There's obviously a hierarchy with a rank structure, and where each crew member has a lot of responsibility. One of the recurring themes I have encountered in working with the U.S. Navy has been the number of Navy personnel who were inspired to join because of science fiction in general, and Star Trek in particular.
However, there are differences as well. Whenever I watch Star Trek, I'm always impressed by the amount of space people had to themselves, and the degree of privacy. On an aircraft carrier most officers have at least one roommate, while enlisted personnel live barracks-style. Communal facilities, such as bathrooms and recreation rooms, are the norm. The U.S. Navy works its people a lot harder than Starfleet does as well; 12-16 hour days are the norm, split between standing watch and other duties. One of the things that always amuses me about Star Trek is the implicit idea that you are always either standing watch or off-duty. Doesn't anybody have to fill out the paperwork to make sure that Jonesy gets sent to the class he needs to qualify for his next promotion? It's not that they don't show it, which admittedly would be boring. There is simply no evidence that paperwork exists at all.
It appears that the most likely opponent of the new supercarrier Enterprise will be China. The most consistent opponents of the starship Enterprise are the Klingons and Romulans. Any similarities between the enemies of the two Enterprises?
Well, in the original series, a lot of people drew parallels between the Klingons and the Russians, and the Chinese and the Romulans. That having been said, I'd hesitate to say that China is the "most likely" opponent of the new supercarrier. The U.S. Navy is concerned about China because it is a stressing case, mostly because our commitments to our allies in the Pacific mean that any conflict with China will be a home game for them and an away game for us. But if you look at how CVNs have historically been used, you'll see that they got a lot of use conducting combat operations without ever having to fight "the big one." Just in the last two decades, you have Iraq (a couple of times), the Balkans, and Afghanistan, plus humanitarian assistance and disaster relief ops in Haiti, Indonesia, Pakistan, and a host of other places. Carriers are just useful things to have whenever you might want to have an airfield nearby but don't happen to have one.