So, cruise line transfers are basically transportation services that are provided by the cruise line company to take you from the airport to the port (and vice versa) for your cruise. These transfers are can be added as part of the price of your cruise and you'll be riding with other passengers who are also going on the same cruise as you. They usually use big buses or coaches to transport everyone.
Private transfers, on the other hand, are when you book your own transportation from the airport to the port with an independent company or individual. It is recommended to book this in advance, and it can be customized to your specific needs. You can choose what type of vehicle you want, like a fancy car or a limo, and it'll be just you (and your group) riding in it. This option is more flexible and personalized, but it can also be more expensive than cruise line transfers especially on the day the cruise embarks or disembark.
If you plan to go early, it will be less expensive to go through private transfers from your hotel to the pier especially if it is a faraway pier from the city center such as Rome airport/city center to the Civitavecchia pier or London Heathrow to Southampton or Anchorage to Seward.
Rome hotel to Civitavecchia pier - Claudio Caponera LimoinRome
PRO TIP: Hotels in the United States provide a cruise and stay package that allows you to leave your vehicle parked at the hotel for the duration of your week-long cruise, provided that you stay at the hotel for a couple of nights before the departure. This deal gets booked fast so reserve early.