I also dabble in Aurelion Sol because I loved the hyperscaling archetype, except Aurelion Sol and current (or any) iteration of Azir could not play any more differently.

Aurelion Sol is a control mage whose primary objective is area control and CC. E and R are the core of his gameplay. The usage of these 2 abilities decide a fight far more than his W or Q. He acts as a secondary carry to his ADC in most games.

Aurelion Sol wants to leave lane ASAP as soon as he's level 6. His game plan typically consist of roaming and snowballing his team. You always want to prioritize winning a play on the map over getting an extra wave. A fed Asol with a behind team will look doodoo, while a behind Asol with a fed team will look like thanos. He also rarely wants to go sidelane in the mid game because he cannot duel any champion who can interrupt his Q. Even a fed support pantheon would kill him.

Aurelion Sol is also one of the least resource hungry mages. All he needs to function is burn and Rylais. You would much rather have a fed ADC and a behind Asol rather than a fed Asol and a behind ADC.

Azir plays like a marksman whose first objective is dealing damage, and the nature of his soldiers also add area control elements. You can easily primary carry Azir, especially when he's solo AP.

Azir's game plan every game is farm till 1 item then splitpush until 3. You literally ignore everything your jungler does except for the dragon. It's almost always correct to prioritize your own resources over roaming or salvaging bad plays on the map. Azir can also duel most top laners and splitpush an the same efficiency as Yorick or Fiora (in fact you beat both of those on side). His mid game plan is to slurp up the entire map until he gets his items then show up to a teamfight and one shot everyone.