There is a hair difference but in practice they are identical to the photographers. The ASA uses an arithmetic scale whereas the ISO uses both arithmetic and logarithmic scales. The ISO also brings several older standards (used in different countries) into one universal standardization.
The ISO and ASA are used for film speeds. Modern digital camera's ISO settings are somewhat arbitrary and are rated comparing to the film of the same EI rating at the same exposure.
In practice, one does not need to know the minute details of how one determines or calculates ISO. It is more practical to know how to optimize exposure and minimize noise. This will require a different set of knowledge, how to determine the right exposure to achieve a desired effect, picking the right aperture, shutter speed and exposure compensation, use/manipulate lighting, etc.