If we're comparing both the Bose and the Dre's i can provide somewhat of a similar comparison. I have 3 of the Beats by Dre Studio headphones (2 real ones and 1 fake one i bought on a business trip to China). I also have the Bose QC15's which are similar to the QC3's. FWIW, i think i am a medium level audiophile. I have about 15k invested in my HT system and i've spent a total of about 5k for the audio systems in both my cars, not to mention my previous cars. Anyways, in my opinion when you compare the 2 they each have their advantages and drawbacks.
Noise Cancelling:
The Bose has by far the best noise-cancelling. Its not even close. I switched my TV to a static channel and turned up the volume. The Bose was amazing. It was almost like putting earplugs on. The Beats did cancel the noise, but not nearly as much as the Bose did. Bose really has the upper hand from this aspect.
Audio Quality:
Now in regards to the audio quality, i think it all depends on how you like your music played and what type of music it is. If you like it loud and rich sounding, then i think the Beats wins this one. The Beats just sounds much fuller, richer, and sharper than the Bose. When played loud there is absolutely no distortion at all. The Bose on the other hand, sounded a little flat and "hollow" in my opinion. Bass response was weaker...but decent. It just didn't sound as rich or full as the Beats. If you like rock, rap, pop, hip-hop or any music that's played loud, then i think the Beats is best. If you like calmer music like classical, or jazz, or acoustical music, then you probably won't tell the difference so much.
Appearance/build quality:
I'm an engineer working in the quality department of a Fortune 500 company. I think i can spot good quality when i see it. When comparing these 2 i think they are about equal with a slight advantage to the Bose in terms of quality. The reason for the advantage to the bose is the hinge on the Beats concerns me. The hinge seems a little weak and i am wondering how it would hold up long term wise. In terms of materials, i think both use good materials and are both equally comfortable to wear. In regards to looks, i gotta hand it to the Beats. These things are eye-catching and scream "bling" and i see many people wearing these things like an accessory. Whether you are wearing them on your head or they are draped around your neck, they are just nice to look at. The Bose headphones look like they could have been made 15 years ago. The Beats are definitely current and state of the art in appearance.
Design/function:
The Bose is vanilla. Its plain and just works. The best thing about it is that it folds relatively flat. The Beats on the other hand, is the neopolitan. Although it doesn't fold as compact as the Bose, the neat feature is that you can use it with your cell phone. The cord has a small switch to pause the music, and answer a phone call so you don't have to take off your headphones to answer a call. The Beats also have a momentary switch/button on the earpiece that you can press to mute the sound temporarily. For example the stewardess asks you what you want to drink, you just press your finger on it, hold it down and order your drink. When you let go, the music continues. Pretty neat.
If you want class leading noise cancelling, get the Bose. If you want good sound and great looks and functionality, get the Beats. FYI, i have a fake Beats and a real Beats and they are amazingly identical. I paid 45 bucks for the fake ones and 300 for the real ones.