I have heard multiple times in photography, the words Bokeh, and Gaussian Blur. To me, it seems that the words are used almost interchangeably, but in some instances, I have heard them contrasted. What's the difference, and what are the definitions of each of them?
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:
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.
Bokeh means the visual looks of out-of-focus areas of a photograph taken using real world optics. For perfect optics an out of focus point of light (e.g. an out of focus start) would be a perfect disc. Real world optics are not perfect and a single point of light will not show up as a perfect disc in the photograph.
Gaussian blur is a digital filter that is easy to compute and looks somewhat similar to out of focus image. However, Gaussian blur does not output a disc for a single point of light in the input but instead a blurred blob with no distinct border.
Here's a visualization of the difference (created with Gimp):
Out of focus can be emulated with digital filters, too. Technically this is called convolution and it's reverse is deconvolution. There exists even algorithms called blind deconvolution where a piece of computer software first computes deconvolution filter and the applies the filter. And the near-magical part is that the deconvolution filter computed this way can remove camera shake and re-focus incorrectly focused image -- to a degree. The process is seriously limited by the noise from digitizer.
The case where digital post-production cannot match real optics is if the original scene has high dynamic range the and the image going into post-production does not contain enough dynamic range. This is because very bright light spot should create bright disc if it's out of focus. However, with limited range for input image, an out of focus filter cannot create the missing original intensity for the disc and as a result the disc will look very dull compared to real image taken with real (high quality) optics.
If one could have low noise and high dynamic range image sensor with a small lens (e.g. a smart phone), it would be possible to emulate current state of art high quality lens with a big sensor. Unfortunately, we currently do not have any technology to create an imaging sensor that has low noise and high dynamic range combined with high resolution and the last part is required for small lens.