In strict and most common usage, there is no real difference between a hot tub and Jacuzzi®. Both are used to describe tubs of hot water which use jets of forced air to produce currents and bubbles, either for therapeutic or strictly pleasurable purposes. Jacuzzi® is a brand name, and so strictly speaking, it describes only those hot tubs and spas manufactured by the Jacuzzi company.
The Jacuzzi brothers immigrated to California from Italy in the early half of the 20th century, and according to the company's website were prolific inventors. Starting with aviation inventions, the Jacuzzi brothers moved into hydraulics, making great strides with the agricultural pump. In 1956 they invented a hydrotherapy pump for personal use. This pump, the J-300, was then sold to hospitals and schools.
In 1968, Roy Jacuzzi created the first whirlpool spa, which he called the Roman. The Roman was based on the hydrotherapeutic pumps of Roy's parents and grandparents, integrating them seamlessly into a standalone unit. Roy brought his invention around the country, capitalizing on a national obsession with cleanliness. It became an overnight success — to such a point that within a few short decades, the brand name had become virtually synonymous with the invention itself.
In colloquial use, a hot tub is often seen as distinguished from a Jacuzzi® or spa by its lack of jets. In this sense, any hot bath or basin of water could be correctly referred to as a hot tub. More often, however, it is used specifically when discussing such things as wooden barrel hot tubs, often with wood-burning stoves or other alternative forms of heating.
This distinction does not hold up in industry discussions, and from the perspective of a strict definition, it is best to treat hot tub and Jacuzzi® as mutually interchangeable words. Product literature, for example, refers often to "Jacuzzi hot tubs," giving the general product (hot tub) and the brand name. If the definition of hot tub specifically refers to those tubs without jets, then the term "Jacuzzi hot tub" becomes an oxymoron. It might be said instead that the term "hot tub" can refer to any tub that contains hot water, while a Jacuzzi® or spa refers to those hot tubs with jets.