In an image that is out-of-focus on purpose, e.g. on a night photo of a street, why do the spotlights look so much different than the usual focus blur?
Such spots tend to have sharper edges/shapes (depending on the bokeh) that are brighter on the border than on the center, while ordinary focus blur looks more like a gaussian blur.
Each point of light from behind or in front of the plane of focus will be rendered as discs in the image. Most of the time the relatively smooth brightness transitions in the background cause these discs overlap to the extent that they appear as a smooth blur.
Point light sources represent strong brightness transitions, so the circle shows up very prominently. The hard edge of these discs is most likely due to image sharpening algorithms that look for edges and increase the contrast either side of the edge to increase the apparent sharpness.
It is also possible that the discs themselves are brighter toward the edge than in the centre. This is the result of over-corrected spherical aberration, and is a common issue (or rather design compromise) with sharp wide aperture lenses. However is often visible without point light sources and gives rise to "nervous bokeh" not the Gaussian-blur like backgrounds you report.
They stand out because they're so much brighter than the surrounding areas.
The out-of-focus blur is not gaussian; every out-of-focus point in the scene projects a fairly evenly-shaded disc, like what you're seeing, onto the sensor. These discs overlap and additively blend so that, in areas without much contrast, the result is a more-or-less smooth blur. But light sources can be much brighter than their surroundings, so the discs they project contribute almost all of the light hitting any given pixel in those areas, and thus stand out.
The reason the borders of the discs are often brighter than the centers has to do with spherical aberration. Lenses vary greatly in how they handle this optical distortion; it is closely related to how smooth and "creamy" bokeh they produce.
