Oxygenating your wort using a tank is leaps and bounds more efficient (as well as more expensive) than agitation or splashing. Simply agitating or splashing your wort to oxygenate it will work for most average gravity beers (you're aiming for a minimum of eight parts per million of oxygen minimum), but will otherwise require a significant amount of effort if you ever need to oxygenate above eight parts per million (PPM), up to 10PPM for high gravity beers. 10PPM of oxygen would be near impossible to accomplish via splashing, since the air we breathe only contains up to 21% oxygen in it.
Using a air pump like what you would see in aquarium tanks should also be used with a High Efficiency Particulate Air (HEPA) filter to prevent any of the ambient microbial contaminants from flowing through the pump, since it is just pulling in the air around it. Also, the stones used on aquarium pumps aren't as efficient as the 0.5 micron diffusers you could get for pure oxygen tanks (plus aquarium stones fall apart quickly and leave particulates in the yeast cake). You could likely achieve the necessary 8 - 10 PPM using the stones for aquarium pumps, but it will take between 15 - 30 minutes to accomplish, where pure oxygen with a 0.5 micron diffusion stone will take only a minute or two.
If you were to taste a beer that lacked the proper levels of oxygen prior to fermentation versus a beer that was properly oxygenated, you will most certainly be able to taste the difference. The most obvious will be that the higher final gravity between the two beers. The one that lacked oxygen will have a much higher FG than the one that was properly oxygenated, the effects multiplied the higher the OG is. That said, it's been my personal experience that over-oxygenating your wort will result in a lower than anticipated FG. I recall splitting a batch for experimentation, the OG was approximately 1.068, one I aerated through agitation by setting the carboy on its side and rolling back and forth vigorously for 5 minutes. The other batch I aerated with pure oxygen for about 5-7 minutes before my tank peetered out on me, and the latter batch finished a full 10-11 gravity points lower than the one aerated through agitation (1.011 vs 1.000).
Also, you would not need to worry about filtering bottled oxygen, no contaminants would survive a can of pure, pressurized oxygen. Also, make sure there's no open flames or sparks near the wort while oxygenating using pure oxygen, as there are few gases that are readily accessible and more flammable than pure oxygen.