I've got both - the Vornado is quieter and more powerful.
The Dyson sounds quieter if you hear it in a public place because its motor produces a higher pitched noise (like a small vacuum cleaner) whereas the Vornado sounds more like a normal fan.
The Vornado is very powerful, I have mine at the other end of the room and I've never used the max setting. The Dyson is less powerful, but it oscillates - the Vornado is meant to be used differently. You place it in a fixed position and it kind of moves all the air in the room. It's really meant to be used near a window or air-con to circulate cool air. The main draw of the Dyson is the absence of 'buffeting', as in you just get a very steady stream of air without 'pulses', but that's moot if you turn on the oscillation anyway.
The other key difference is that the Vornado is made in the US and comes with a 5-year warranty whereas the Dyson is made in Malaysia I think, with a 2-year warranty. I guess that's the reason for the high prices - the Vornado is supposed to have an industrial strength, heavy-duty motor whereas the Dyson is... unique and very easy to clean. With the Vornado, you still have to dust the blades and wipe the grille, whereas the Dyson has tiny air intake grooves so it doesn't really get dirty.