Anivia peels herself better than Sol, due to her Ult's innate slow, her wall being able to block opponents off, and how she can instantly reactivate her Q. Sol has a delay with his Q, so assassins destroy him. Sol is better at chasing however; More sustained damage over the burst damage from Anivia's E + R/Q. He's harder to do well with.
Azir's comboes aren't that hard; Just read a guide online or watch a youtube video. The mastering of the comboes is as hard as it was to learn old Alistar's WQ (before they made it easier). Sol, you have to learn through gameplay; when to gank, when to engage, when to disengage, due to him not having any escapes, or peel, as well as having little burst except for his ult. Stutter stepping is not much difficult either. Aurelion Sol also has to "stutterstep" (his is somewhat unique, basically he has to keep a certain distance from his target at all times to optimize his W/Passive ranges without being in range to get obliterated. More often than not, Azir uses his ult to shove the opponent into his allied team to get destroyed, while he stays safe behind the wall, using the soldiers for the AoE Dps. Once Sol uses his ult, he has no escape; Azir has E. Sol doesn't have mana regen in his core build as well; Rushing Rylais. While you can build RoA on Sol, you also do it on Azir.
Sol has the advanatage of being a new champ. Even as a new champ, he hasn't been really popular, as he doesn't have the "broken new champ" thing champs like Ekko had. I have yet to see him in ranked.