A "car transport agency" moves your vehicle for you from one location to another, for example, if you purchased a classic car at an auction in Scottsdale, Arizona, and you want it moved to your home in New York State. The best such agencies will load your vehicle into an insulated, climate-controlled enclosed trailer and have a truck haul it to its destination, so that it will neither incur additional mileage nor be damaged or weather-worn in transit.
The next level of service by a car transport agency is to load your car onto a rented, open trailer, cover it with a tarp, and tow the trailer and car using another car or pickup truck wherever you wish.
The cheapest such "agencies," if they can even be called that, simply hire a licensed young person to drive your car to its destination. I believe this is the case in the example you offered.
A "car or truck rental company," like U-Haul, Hertz or Avis, provides you with a vehicle to move yourself or your belongings from one place to another. Some people prefer to rent a car for an extended trip to avoid racking up miles on their personal vehicle, and they return the car to the same rental agency location where they picked it up.
A "moving company" (such as Bekins or United) is generally a company that ships and/or stores your household belongings when you are moving from one part of the country to another. They may hold your belongings until they have a full truck load if you are moving from a relatively small city to another that is many hundreds or thousands of miles away. They will also store your belongings in climate-controlled warehouses if, for instance, you accept a job overseas for a year or two. Moving companies will come to your home and pack your furniture and belongings for you, relieving you of the responsibility of packing things up yourself. A thousand-mile move of a household full of belongings can easily cost upwards of $10,000, although if you are moving because your employer has transferred you to another city, the employer will often cover this cost. Once your belongings arrive at your new home, the moving company will also unpack everything, placing furniture and other items where you wish them to be, and take away all of the boxes, blankets and other packing materials.
A moving company is more expensive than renting a U-Haul or Penske truck and doing it yourself, but it is also much more convenient.