OK, so I have an HTC One, which has a Qualcomm dual core S4 SOC. There is an international version of the same phone with a quad core in it. Same goes for the Galaxy S3.
The main drawbacks on the S4, being dual core, is that in HTC Sense image data can't be offloaded onto the extra 2 cores, so that some images don't load in the interface until I'm fully in that area. For example, when in the application list in my launcher, if I drag from "All" to "Frequent", I don't see any Icons in the "Frequent" area, just a grey screen that says "Frequent" until I let go. No preview for me, since I'm dual core. Similarly, I get no image preview when in my Contact list, moving from, say, Groups to People..... And, a another minor negative, since the background can't be offloaded, Live Backgrounds are more likely to cause interface lag, or even cause minor breaks in sound when listening to music. As the background ends up loading one of the 2 cores, it causes the sound to break for a split second, and hence the hickup. Doesn't happen on the quad core.
Is this a big deal? Nope. Would I rather have the same phone with a quad core in it? Absolutely.
Qualcomm certainly wouldn't be saying anything negative about more cores, if they weren't behind. They're making Quad Core chips too. I bet their stance on the matter changes soon, when their quad core Krait chip is found in phones in larger markets. Then suddenly it'll be "Check out how amazing our quad core SOC is, and how much better it performs than dual core chips"