For the purposes of this analysis, I'll assume that you build Frozen Heart and Rod of Ages on Ryze, in addition to the Mana Charge item. This would cost 7900 to 8500 gold. (Another common Mana item built by Ryzes is Banshee's Veil, increasing the cost to 10400 to 11000 gold.)
With these items and the Expanded Mind mastery, Ryze will have 3362 total Mana at level 18. Assuming a standard 9/0/21 AP mastery page, you'll have 98 AP.
Let's do some calculations now.
With a Seraph's Embrace, your AP will change to
262.1. Q (Overload) will do
490 damage, W (Rune Prison) will do
513 damage, and E (Spell Flux) will do
256 damage per bounce. There are two common Ryze spell rotations: QWEQ and QWQEQRQ. The first will deal
2007 (unmitigated, i.e. assuming 0 MR) damage to a target if his E hits twice. The second will do
2988 damage to each target that E hits twice.
With a Muramana, your AP will stay as
98. However, you'll do bonus damage on each spell. These calculations are a little more complex, but they come out to
2247 damage on the first rotation and
3433 damage on the second rotation. In addition, Muramana is 600 gold cheaper. Note, however, that to simplify things somewhat, this ignores mana regeneration, meaning that you actually deal slightly (though probably negligibly) more damage than these numbers indicate.
Therefore, from a pure damage standpoint, Muramana is slightly superior, though as hammythepig mentioned, Seraph's Embrace has an extremely attractive defensive component.
3.5 Balance Update: Now that Muramana does physical damage instead of magic damage, it becomes slightly less viable on Ryze, since it no longer benefits from any magic penetration. On the other hand, it allows for strong mixed damage from Ryze, which can be useful in AP-heavy team compositions.