The video speaks for itself, but I wanted to demonstrate the fairly minimal benefit that you get from turning software video stabilisation on in capture on most smartphones of today. The Nokia 808 stands in as a fairly typical example, though it's true that if I'd have used an iPhone 4S or 5, there would have been slightly better results, given the latter's accelerometer-based stabilisation. Still, my conclusion remains that for general use, there's little benefit in software stabilisation - and in some cases the video footage can be made worse ('tearing', etc.)
In contrast, one of the Nokia Lumia 920's party pieces is the (hardware) Optical Image Stabilisation (OIS), which acts as a steadicam within the phone's body, with dramatically smoother results, as you'll see in the video below.
Although the Nokia 808 comes off rather worse in this comparison, it's worth remembering that it does have other tricks up its sleeve when capturing video, not least the PureView lossless zoom and full stereo HAAC audio capture. Plus the two devices also have very different strengths for stills capture too. Horses for courses, etc.
NB. The sequences were shot in succession rather than simultaneously, for practical reasons, in case you wonder about mismatches in passing cars, pedestrians, etc.!!