By the time you are on the train it’s past 10:30pm. Presumably you have luggage, you will probably be tired if you are coming from across some Ocean, and your hotel may be a few hundred yards from the station. Also the Termini station and immediate surroundings at night, although not unsafe, has some seedy people hanging around in the evenings. Also the LE train is 28€ for you and your wife.
Because of all these factors, the taxi (48€ flat fee), may be a wise investment. It will drop you off in front of the hotel of your choice. No tip required (or even advised) to Italian taxi drivers, all of whom are proprietors of the taxi and none of whom are poor.
It’s your choice. The train+a little walk saves you 20€.
The Hilton at the airport would be a great choice, but it’s 191€ for the night you arrive, therefore not a budget hotel (but not extraordinarily expensive either for what you get). If it’s too expensive for your budget don’t bother with other choices nearby, because there are none. All others are at least 15€ or 20€ away by taxi. Just go to near Termini station, that way in the morning you are one step ahead on your way to Naples. Otherwise you spend extra taxi money for nothing.
If you want a nice reasonable hotel just a few blocks from Termini, I’ve sometimes stayed here when I needed to be near the station. They have a nice bar too for a night cap before going to bed. Their rate for 2/18 is 117€ (non refundable) or 133€ (with free cancellation).