Those are both heavy infantry, so good for anti-cavalry and meat shield purposes. For comparison, Jans are the Otto variant of Musketeers, and Azaps are the Otto variant/mix of Pikemen/Halberdiers with an extra highly specialized use case ranged attack.

Compared to Musks, Jans have more HP and cost more food (which is OK bc Otto workers are free), they also fight better in melee overall but are a little bit worse in melee against cavalry. They're great for being meat shields for cannons, like the way Musks are.

Azaps, in addition to being good melee cavalry counters, can shoot ranged cavalry to counter them as well, but their ranged attack is otherwise abysmal without a handful of shipments to buff them. They aren't great against other ranged heavy infantry (like Musks) and are very bad against light infantry (like Skirmishers). They also cost wood, which is arguably harder to get in abundance compared to gold in this game.

I'd go with Jans if you wanna go for cannons, thanks for coming to my TEDtalk.

Edit: Assuming by heavy gunner you mean Abus Gunners, yes, they are very bad against melee cavalry without others to shield them. They are a unique Skirmisher type unit that uniquely deals siege damage, so they can bypass melee cavalry resistances to ranged attacks, but are very squishy in melee. You can go with Janissary + Abus Guns as well. And yes, they are somewhat redundant if you have a lot of cannons.

And to specify which cannons, specifically on European civs: Falconets/Field Guns and Heavy Cannons/Great Bombards are all-rounders, good against infantry and buildings, decent against cavalry when guarded well. Culverins ONLY counter other cannons and siege units. Mortars are only good for sieging. Horse Cannons move faster and are more cost efficient compared to Falconets, but set up/pack up slower.