On the quest side, probably the strongest argument is that it just feels right that you have to go through with some epic tasks to truly earn your legendary weapon, such as defeating the master of the Liches and an Old God to reconstruct Medivh's staff, or summoning and binding an Elemental prince to obtain the Thunderfury. Because you engaged in a legendary storyline and a legendary undertaking, you have earned the right to wield a powerful weapon.
There's a few answers to this idea that the BC-era Legendaries can provide.
First, sometimes just killing the boss is epic enough. After all, Kil'jaedan, who is one of the most powerful beings in existence, was moments away from entering our world completely when you pushed him back. It seems that's an epic enough feat to deserve a bow.
Secondly, sometimes a quest just doesn't seem needful. After all, Illidan's blades are Illidan's blades. It doesn't seem like he keeps them at low power or let them leave his hands -- and thus it makes sense that they'd be ready to use when you them from him. Sure, you could technically make up a quest where they're drained of power and you need Archimonde's essence, one of Kael'thas' orbs, and 30 demonic sharpening stones off Black Temple bosses to make them wieldable again, but really, there is a point at which those types of quests can feel a bit arbitrary.
Thirdly, quests can often wreak havoc with raiding schedules. It happened quite often with Atiesh. I knew many raid groups that just didn't want to kill C'thun anymore after Naxxramas came out. Getting 40 people to dodge Dark Glare and kill tentacles was just too hard for them, or they didn't want fight through all the annoying trash, or they didn't really need any loot off the other bosses... the list went on. Of course, the side effect of this was that none of their casters could hope to see Atiesh, since C'thun held the base of the staff. Dropped Legendaries mean you never have to convince your raid group to go back and pick up old content again.
The other major argument I've heard in support of questing is that it supposedly removes a little bit of the RNG and gives you hope that you can obtain a legendary without waiting forever and hoping that Illidan drops a glaive instead of a blindfold this time. Having the focus of a quest to finish both gives you a clear goal to work for besides killing the same mob over and over and hoping for a drop.
Of course, the problem with this is that the quests for the Legendaries still depend largely on RNG. I knew many a guild who cursed every time they ran Molten Core and never saw one or the other of the Bindings of the Windseeker or an Eye of Sulfuras drop. The Splinters of Atiesh are not a guaranteed drop either, so you easily might have to kill a few hundred bosses in Naxxramas before getting enough splinters to form a Frame of Atiesh -- and that's assuming your guild or raid group is kind enough to default every single splinter to you. You can't really escape the RNG, or else you'd have every other Rogue, Warrior, or Paladin who ever set foot in Molten Core and saw Ragnaros die carrying around a Thunderfury or a Sulfuras -- and then those weapons probably wouldn't feel that legendary, honestly.
So really, in the end, it feels like the idea of droppable Legendaries just have a lot more going for them. Still, I can't help but think that I'd like to see a few more quested Legendaries, if only because there is a good sense of accomplishment and awe when you finish a well crafted epic quest line. I know this by personal experience, as one of my proudest MMO memories is still completing a long, story-driven quest line in Everquest for a ancient sword known as the Greenmist. I'd love to recapture that feeling in WoW some time. Perhaps the best balance is to take it on a case by case balance. For incredibly epic bosses, it makes sense have a Legendary be a simple drop, while for lower level raid dungeons, you can get away with a quest to create a little bit of extra work for the player and to give a different sense of accomplishment.
We'll have to see what Blizzard does with Legendaries in WoTLK. It's wide open right now. While we know that they want to bring back Atiesh, it might be a bit much to expect it in the upgraded Naxxramas, due to Karazhan and C'thun being obsolete, so we might have to wait for another opportune moment, such as if they implement the Karazhan Crypts. We know from the last Blizzcon that we may not be able to wield the Ashbringer either. Still, perhaps we'll be able to cleanse another corrupted Runeblade, and I'm still pulling for a Legendary rifle from Uldum or Ulduar or Thor Modan, if only because Dwarves with guns is one of the best fantasy concepts ever invented, and it pains me that my Dwarf Hunter must currently wield a crossbow.