This guide is pretty decent at explaining general concepts, but is unfortunately sort of shallow where Noct is concerned.
Diurnal is probably easier to get a feel for healing with, but Noct is kind of overpowered right now for raid stuff.
If you're Diurnal, you can cast Aspected Benefic on him he gets healed for 200 potency now, and then over the next 18 seconds gets another 840 potency healed via the regen effect. This means one single cast gets you 1040 potency of healing, but it takes some time to see the full impact. It will get his health up in times of constant, low damage. If your Aspected Benefic heals for 2,000 HP, then every three seconds another 1,400 HP will get restored to the tank for the next 18 seconds.
If you're Noct, you cast Aspected Benefic and he gets healed for 200 potency now, and gets a shield worth two and a half times the heal. So 200 potency healed to his HP, and then 500 potency in a shield. That means if your Aspected Benefic heals him for 2,000 HP, he's also going to get a shield that lasts 30 seconds that will absorb the next 5,000 damage he takes without touching HP at all.
The Noct shield is lower potency, but it's available immediately. If the tank is taking heavy damage, this can be a big difference since you can keep putting shields on him back-to-back and getting full value out of the cast without needing 18 seconds to pass in between (usually you'd want to do the healing heavy lifting with other spells, but since Aspected Benefic is instant-cast it's nice to be able to toss out while moving or whatever.) It also lasts 30 seconds, so if he takes no damage for the first 15 seconds and then a bunch of damage in the last 15, you're still getting full value out of the shield. Vs a regen where if he takes no damage for the first 9 seconds, half the regen ends up being wasted. This also means you can drop a Noct shield on a tank before he starts the pull, and it'll eat a bunch of the damage during the pull while not generating any threat, which can be nice.
Either works for dungeon content, and either works for raid content but you usually will in 8-man content pick your sect based on the other healer. If you get paired with a WHM, pick Noct so you can put up shields before heavy-hitting attacks while the WHM uses their regens to gradually bring party health back up. If you get paired with a SCH, go Diurnal so you can put down regens while the SCH is proactively shielding. If you get paired with another AST... just kind of see if you both have a preference.
Okay, that makes sense to me. So I think my only other question about Noct which I think I know the answer to but I'll ask anyways; does that 2.5x shield count for the number that the heal is for and not how much it actually heals on the target? For example like healing someone at full health will still give them a shield, right?