Bokeh is specifically the out-of-focus areas of an image. Gaussian blur is an algorithm to fog selected image areas, to hide details or make them look out of focus.
The main differences:
- bokeh is created optically, gaussian blur in post-production;
- in bokeh, the amount of how wide an out-of-focus point will be smeared is determined by its relative distance from focal plane, whereas gaussian blur is applied to a two-dimensional image where no distance information is present, thus all points are smeared equally;
- in bokeh, the smearing characteristics depend on configuration and aperture shape of the lens, whereas gaussian blur is always smooth;
- a small light source will be rendered as an aperture-shaped figure with quite well-defined edges in bokeh; but gaussian blur renders it as a spot with fading edges;
- in bokeh, noise is present at the same level as in in-focus parts of image with same luminance; gaussian blur kills noise, so there'll be less noise than in non-blurred parts of image;
- in bokeh, light areas will dominate over dark ones, while gaussian blur gives preserves the ratio of dark-light areas.
A sign in a train station, taken with f/10 (giving deep depth of field).
Gaussian blur performed on background parts of the previous image.
A sign in a train station, taken with f/2.8 (giving shallow depth of field and natural bokeh).
So, all in all, you can use one to fake another, but the result will be similar only for low-noise bokeh containing items on roughly a plane parallel to focal plane, not including any significantly lighter areas or light sources, and taken with a lens that has a smooth bokeh.