The difference will be in the walls. A hard-shell hot tub will have hard sided walls that support the water inside. They will not need to be inflated to setup and will likely have extra features like built-in seating. Hard-shell hot tubs can also be placed in colder locations and may be more durable for outdoor use.
A 2 person inflatable hot tub will have soft sided walls that need to be pumped with air to build the structure. An inflatable hot tub will be easier to move, as well as simple to deflate and store. They will have a strong material that is resistant to tears or punctures to ensure all your water stays inside.