The limited sensor and lens size of a smartphone's camera places a limit on image quality. That said, most people only take small images for FaceBook, Instagram or the phone's screen - most smartphones are more than enough for such tasks. While generally good enough, image quality varies a lot and perhaps the worst measure is pixel count - the number of megapixels.
Here is a camera comparison test of one of the most popular smartphones and another that aces the megapixel race. Having played with both cameras I'm not terribly surprised.
A bottom line is that if photography is important to you, get a dedicated camera. A $200 point and shoot will run rings around any smartphone camera and the difference is astonishing if you move above $500.
That said - having a camera everywhere you go is a feature.
Although there are severe physics-based limitations, smartphone photography is good enough for many (most?) people. Nokia probably has the best camera, although its ease of use is reportedly poor and the its phone probably will never sell in quantity for other reasons. The most important curve to follow is probably the iPhone as it is the most common. A usage test of photography with the iPhone 5s..
Of course the photographer is much more important than the camera, but are we at the point where casual photographers can get as much out of a smartphone as they can with a dedicated camera? Zero incrementla cost, small,, always with you and having a persistent connection to the Internet are important elements ... and it is probably just as important for camcorders.
High speed still photography is within reach of the amateur - you just have to be a bit clever with flash illumination and triggering the photo. Video slow motion, at least beyond 120fps or so, is more difficult and expensive.
Smartphone cameras are getting better, but even the best are not even close to a good pocket camera - basic physics makes sure of that.
In a NY Times review David Pogue looks at the Sony QX-100 - a novel design that squeezes most of a camera's electronics into a good lens and communicates wirelessly with your smartphone to use it as a viewfinder, interface and connection to the Internet. He agrees with all of the other reviews I've seen .. the photos are far better than anything any smartphone can deliver, but the user experience is not good (some reviews would say terrible).
Of course this won't matter to most people. Smartphones are giving them "good enough" shots for Facebook and Instagram as well as their own library and they are reasonably easy to use. If you want to shoot in low light, get high quality images and become creative you simply need a "real" camera. The best smartphone camera is in the Nokia Lumia 1020, but it compromises the rest of the phone, which has serious additional problems (one reviewer suggested it is for people who take a lot of photos and need more quality than a regular smartphone, but not that of a $150 pocket camera - oh .. and the only thing they use the smartphone for is texting and voice)
But give it time ... perhaps this concept with work itself out. I can see kids getting interested in a decoupled camera/phone.
The Bigshot camera - designed for kids, but perhaps useful beyond that. Learn about some of the fundamental elements associated with cameras. (check out the learn section). The cameras in their phones are probably better, but this might be a good teaching tool.
And then there is the issue of learning photography ...