I've always been a huge fan of blurry photos, I guess in part because almost any camera is able to take blurry photos, though some better than others.
What's the difference between real-time and post-production blurs?
by Paul Cezanne
In general: "real" blur, either due to optical characteristics (including depth of field, chromatic aberation, spherical aberation, and more) or due to movement, is based on more information. It includes the three-dimensional and time aspects of the scene, and the different reflection and refraction of different wavelengths of light.
In post-processing, there's only a flat, projected rendering to work with. Smart algorithms can try to figure out what was going on and simulate the effect, but they're always at a disadvantage. It's hard to know if something is small because it's far away or because it's just tiny to start with, or if something was moving or just naturally fuzzy — or which direction and how quickly. If you're directing the blur process by hand as an artistic work, you'll get better results because you can apply your own knowledge and scene recognition engine (in, you know, your brain), but even then, it's a lot of work and you'll to approximate distance and differing motion for different objects in the scene — or intentionally start with a photograph where these things are simple.
In the World of Tomorrow, cameras will gather much more information in both time and space. The current Lytro camera is a toy preview of this. With a better 3D model, the effects of different optical configurations can be better simulated — and of course motion blur can be constructed from a recording over time.
Both motion and out-of-focus blurs can be simulated in software given enough time but the results are rarely as satisfying as the real thing. Faking it works best when you have only two levels of blur, for example a single moving object or a single subject with the background a fixed distance away.
To really simulate lens blur the blur radius varies proportionally with distance from the focal plane so you would have to map out the distance to every point in the image. If you have a flat background of equal distance such as a wall then the problem is reduced to that of cutting out your subject, which is more manageable). One thing you can often get away with is taking an existing blurry background and making it more blurred.
Unless you spend a long time on it you wont fool another photographer, though you will a casual observer.
Motion blur is slightly easier provided the motion is side to side, as it reduces to a problem of segmenting independently moving objects (cutting out your subject). What you may have to do however is fill in some extra background either side of your subject as in real life you see slightly move of the background than you would if the subject were stationary. This is because background is revealed as the subject moves across the frame.