Am I really going to chime in here? I'm a complete newbie to homebrewing, so likely someone more experienced than I will come along and in true forum fashion, kindly and politely tell me I'm completely wrong!
That said...
To my understanding, stirring vigorously and an air pump should both aerate the wort equivalently, provided, of course, that one stirs long enough, vigorously enough, and thoroughly enough.
I notice that some homebrewers simply use an aquarium air pump. Aquariums I'm far more familiar with. Yes, people use those pumps to aerate and oxygenate the water for their fish, but they don't work the way many people think they do. The oxygen in the bubbles as they rise isn't what's absorbed into the water. Rather, when the bubbles break the surface, they create ripples. These ripples, or surface agitation, creates friction with the air and promotes gas exchange, including oxygenation. Again, the oxygenation happens at the surface, not from the rising bubbles. This is why in an under oxygenated aquarium one often sees the fish hanging out near the top where the water is more oxygen rich.
Bubblers like this for aesthetic reasons are still popular and in use, but in terms of aeration, everything I've read on aquariums says that just about any modern aquarium filter that pumps water out near the surface creating this surface agitation is all the more anyone needs.
One important thing to note: modern aquarium filters draw water from an intake closer to the bottom and exhaust it near the top. This is important. If there isn't the bottom-to-top movement of the water, and aeration happens at the surface, the deeper water in the aquarium won't get aerated. So just whisking the top of the wort like crazy, or shaking the vessel and just sloshing the top of the wort around, or using a spoon or other implement that doesn't reach the bottom of the vessel and move that liquid to the top to be oxygenated won't oxygenate the wort near the bottom of the vessel.
So... putting this all together would tell me that vigorous stirring and creating surface agitation over a long enough period of time combined with mixing wort bottom to top should work as well as an air pump (the rising bubbles help encourage the bottom to top movement of the wort).
Of course one could let the bubbles do all the work and relax and have a homebrew instead of stirring like mad, but stirred adequately, I would think the results should be similar.
I don't know if using pure oxygen would saturate the wort with more oxygen than simply using air, but I would suspect that even if one does super-oxygenate the water, it might not stay in solution too long as it would eventually gas off. Long enough to help boost the yeast population perhaps? I haven't a clue.
So there you have it.
And now I await, in true forum fashion, I await the very kindly and politely worded replies letting me know that I don't really know what I'm writing about.