If one were to describe the nature of a beast soul as opposed to the characteristics of a partial transformation, they could only simply say they were quite different from one another.
The partial transformation, the incorporation of the physical traits and abilities of your spirit beast into your body, was a pure physical transformation. It could grant a cultivator an increase in strength as well as the abilities of your spirit beast depending on the type of beast one possessed. But the beast soul, on the other hand, was the combination of your soul with the soul of your spirit beast. It was spiritual in nature, to be specific. And the beast soul, unlike the partial transformation, granted the cultivator a massive increase in your spirit energy as well as certain other aspects.