Akko and Diana have the same goal, they're just going about it in opposite ways.

Akko is following in Chariot's footsteps learning about all the secret stuff of the Academy because she acquired the Gunbuster Shiny Rod, while Diana is taking the academic and methodical approach.

The magical butterflies episode was a good indication of how this is going to end. Akko has the Shiny Rod, enthusiasm, and the willpower to seek out the mysteries of the school and its founding, but she's too stupid to actually pull it off. Diana also secretly shares the same wonderment and awe of magic that Akko has, but is diligent enough to help Akko pull off...whatever this is growing into, a revival of magical power I'd guess.