Math time
Ryze's mana: 1240 at level 18
3 Archangel Staves: 3 x 400 + 1000
Mana from Lichbane: 350
Total mana: 3790
AP from Lichbane: 80
AP from Archangel Staves: 3 x 45 + 9% of mana: 476.1
AP after Rabadon's Deathcap: (476.1 + 140) x 1.3 = 800.93
Overload damage: 160 + 0.4 x AP + 0.065 x mana
= 160 + 320.37 + 246.35
= 726.72 Damage
Rune Prison damage: 200 + 0.6 x AP + 0.045 x mana
= 200 + 480.56 + 170.55
= 851.11 Damage
Spell Flux damage (per bounce): 130 + 0.35 x AP + 0.01 x mana
= 130 + 280.33 + 37.9
= 438.23 (1314.69 If all 3 hit) Damage
Lichbane damage: 106 (base AD) + 800.93 = 906.93 Damage
Full combo damage: 2922.99 (3799.45 If all 3 Spell Flux bounces hit)
If we take normal magic resistance from carries into account (about 30%), the combo still deals a respectable 2000-2600 damage. And considering the cooldown on his skills, it won't take long to dish out another 1000 damage.
This is of course, without any masteries and runes. I have a runepage which gives me 644 mana at lvl 18, which would boost both my AP from AA staves and the pure damage of my spells quite well. And masteries help too.
Oh and the 25% spellvamp works just wonders with this build.
But you're not getting the most out of him if you're building that way.
At that point, you're effective health is about 3300. My effective health becomes about 7100. That's a noticeable difference. This matters because you are a short-ranged mage. You don't get the liberty of skillshots through things, so you will have to be up in someone's face. Your full burst? (Which, by the way, is WQERQ, so you count Q twice.) That has only done half of my health. My full burst does about 3/4 of yours. I heal faster, cast more often, and I also get a nifty spell shield to stop some of those pesky spells. And since I am tanky, I don't need to be behind everyone.
Not only that, but I end up with over 4500 mana. So, I still end with more. And since you're going AP, you're going 21/0/9, I'm assuming, but then you're missing the most crucial parts. You can go 9/0/21 with 0 repercussions. Again, you do slightly less damage, but you more than make up for it with tankiness, which you can use throughout the game. Since you can be in the frontlines, you're not worried as much. And, since I'm going 9/0/21, I get several things that help:
Reduced Summoner Cooldowns. This means if you have an exchange in mid, you will have your summoners before they do. This is important when dueling.
Increased xp. You reach higher levels faster. An extra level matters.
Increased hp regen. There's this little mastery in the Support Tree. It's called Strength of Spirit. You get 1% of your mana in hp regen. That about triples your regen. Even lvl 1 with 577 mana, that's a third of a health pendant. This works really well if you're tanky.
The things I don't have aren't many. I lose a bit of mpen, damage, and AP. Those don't matter to me, since my job is to hold the frontlines, not necessarily to nab all the kills, though I tend to do that anyway.
The difference between my Ryze and yours is my Ryze starts fights. Yours has to end them.
Your combo does about 1.8x my damage, but only because of lich bane. But, again, short range and poor AP scaling does not make him a good lich bane user, especially since you usually are casting too often to auto attack. I also get about 2.2x your tankiness, and if I'm focused, I can still last throughout the fight. WotA and natural hp regen work better with tankiness.
I also don't appreciate the rune choice, but that's a little different. I use a blue one to make my CDR at 40% fully built. This means the blue I would need can go to someone else, like the AD carry.
There's so many little things it adds that are better than increased damage. The only thing I really lose out on is pushing power, which Ryze has only a bit of. However, if your team just doesn't need any of those (because you have Malphite and Galio manning the frontlines or whatever), then your build wouldn't be too bad. I still would rather use that on Karthus, whose sole purpose is to die, but he's much more useful for auto-attacking, since you're not constantly smashing any button but Q. QAQAQA. Works much better.
tl;dr: Ryze is better tanky. He loses damage and gains tankiness, but can start fights and still do enough aoe damage to make the enemy team cry, and they can't insta-gib him.