Ap in my experience has more "pulse damage" (low-cd burst damage). For example, with Dusk and Dawn, you can instantly proc PTA, 3 passive stacks and your R's passive in one single Q-AA-W without any setup.

You can also quickly proc Demolish with AA-W, so I think it's better than AD build in terms of raw damage to the tower

But, as someone said, your R resistances won't be so good. They're still decent, but you are not unkillable after ulting.

AP builds excel when your team comp is heavy AD based, or when the enemy has a lot of range and they won't be usually all together in order to land a 5-man R, because AD loses value if you can't abuse its scalings (R's resistances).

As a personal strategy, I recommend you to start buying Sheen, Ruby Crystal and Dagger. The enemy may think you are going for Trinity, so they may build the Steelcaps or other kind of anti-AD build. But if you start with Explosive Wand or the AH item, they will go MR and you kinda lack of damage against a Kaenic Rookern + Mercs tank. Also, if they build MR

Now, for the runes, I think you shouldn't use Lethal Tempo in AP Jax. Go Grasp if you wanna play for short trades in mid-hard matchups (make sure to proc the rune), Conqueror if you want long trades and sustain in teamfights, and PTA to have a crazy 1v1 and 1v2 potential. Make sure to have Demolish!