Don't get me wrong, I'm not saying that WAR isn't hilariously easy to heal. I do think that ToB skews the actual healing number a little bit in terms of slight inflation, especially if you pop it at full HP prior to taking a buster as I'm fairly certain that HP increase still counts towards your healing total even though you effectively lose that HP that the initial activation granted you after 10 seconds. That aside though, I would argue that while WAR and DRK remove a *comparable* amount of damage from existence, that the edge remains with DRK for one reason; that being that NF is directly tied to the WAR's damage output at any given moment.
Since WAR's rotation is almost entirely burst-oriented, most of its non-IR time is spent on your normal Storm's Path/Eye combos. Since you only use Fell Cleave outside of IR in order to avoid overcap you're more than likely going to have at least 50 gauge to use one when you need to use NF, so for our 'average' Nascent usage we can reliably allot a Fell Cleave, Maim, and Storm's Path, which gives us the following.
Fell Cleave: Averages a 18k hit, 9k healing.
Storm's Path: Averages 11.5k, 5.7k heal on top of the 6.7k heal attached to the ability
Maim: Around 8.8k-ish damage, around 4.5k healing
Auto attacks: Around 3.9k damage, you'll get two during NF, so 4k healing
This totals out to a roughly 29.9k, or 30,000 HP heal if nothing crits. Now, it's been a bit since I calculated the numbers, but if memory serves TBN in current gear shields for around 43k HP. This leaves our average NF usage a whopping 13k health in the red, which can add up quite a bit over an extended period. Now since the WAR rotation typically tries to keep an Infuriate charge ready to go at most points in said rotation, you could potentially dump an IC into NF rather than either that Fell Cleave or, more likely, Maim. This changes our total healing value from 30k to a whopping 52.4k health (IC average of 54k hit with 27k healing), or 9.4k health ahead of TBN. This does, however, present one downside to it; you now no longer have that Infuriate charge to use on IC when the party's raid buffs come back up. With Inner Chaos being such a huge chunk of damage at 920p, this does amount to a moderate DPS loss; a scenario that the DRK does not, in any way, have to entertain with TBN. The DRK also has another slight advantage when considering that, while TBN and NF can be used with relatively equal frequencies, the WAR will most certainly in a few of those NF windows be able to put out nothing but a Heavy Swing, Main, & Storm's Path/Eye, while DRK's mitigation value is consistent across each use.
TL;DR - Both WAR and DRK are easy on their healers, and NF is capable of outright blowing TBN's shield out of the water in some scenarios, in an average use TBN will still come out with a bit of a lead in terms of damage dealt with.