Applications are the hypothetical powers you obtain when you acquire a power. So, if I have Earth Manipulation, I’d most likely gain access to controlling all minerals, crystals, and obviously dirt and rock. Every application for a power is a part of it and obtaining the power will inevitably come with the applications if mastered.
Associations are hypothetical powers you could obtain when you acquire a power. So, if I have Earth Manipulation, I’d get all of the applications, but I could get Gravity Manipulation by controlling the gravitational force of the entire planet. Most associations do not come with the power, however they can be obtained through various means.