Advisory ice detection relies on TAT and visual identification of liquid water. It is up to the crew to take action based on the parameters and observations. This is considered 'the old and reliable' method.
Primary ice detection is automated and relies upon purpose-designed sensors and is a certificated system. These systems are typically quicker and better at detecting icing conditions than the aforementioned Advisory method, often doing so directly.
In reference to the MEL, you can typically go without Primary, but not Advisory Ice Detection equipment as the latter is considered the 'bare bones' fallback.