The graininess in film is related to but different from the "grain" out of which images are said to be made. Graininess refers to a subtle (or even not so subtle) haze or shimmer across an image, whereas grains refer to individual silver clusters or dyes in the film.
Now an image is not really made out of such silver or dye "grains". Rather, it is the other way around, the grains are made out of the image, by which should be understood that the distribution and density of the grains is a function of the image. The image determines this arrangement of grains. The image is encoded in terms of grains, or more specifically, encoded in terms of variations in the grains. It is the variation which encodes the image, rather than any particular grain as such.
During film projection, the image, encoded in terms of this variation in the grains, is decoded. The image is reconstructed. For an ideal reconstruction the material otherwise encoding the image should completely cancel out. To the extent that it doesn't completely do so there will be a residual effect we call graininess or noise. It is the difference between the image and it's reconstruction. The fact that we can appreciate such a difference is quite remarkable given that we don't normally have the original image against which to compare it's reconstruction. If we see graininess at all it is because we are familiar with optical images.
Now the image, as mentioned, is encoded not in terms of grains per se, but the variation in such, which, due to the way in which such grains are prepared (prior to image exposure), resist the image in a random way. Those separable concepts otherwise clear to us at the level of an image (definition, dynamic range, etc) become less clear under the microscope. They become fragmented in one place, and entangled in others. The components of the image becomes unclear. We have to look up from the microscope and stand back to re-appreciate the image. It's a peculiar thing but it demonstrates the idea of the microscopic as a function of the macroscopic, rather than the other way around. It is not the image made out of grains but the grains made out of an image (so to speak).
Now it should be clear by now that the image is neither made out of grains nor made out of graininess. Nor is graininess something added to an image. Graininess is a side effect of the encoding/decoding pipeline. It is noise. It has no structure. It constitutes the absence of a signal. A lack of information. A zero signal. Or better: a null signal.
If we appreciate any graininess in a decoded image it is only because we appreciate the image we've otherwise decoded. The graininess is a result of this decoding. Not an input into such. We speak of seeing through this grain. We can see the image to the extent we can.
The image itself is not in any way altered by this grain. It is for this reason that adding grain to an image has no effect on the image. We see through it.
If we use synthesised grain in certain post processes it's not for an aesthetic purpose but a technical one. An image that is to undergo transfer to a lower bandwidth domain can be pre-filtered using a form of noise that redistributes the energy in such a way that the resulting lower definition signal retains more information than if it did not undergo such a redistribution of energy. Using a random signal ensures there is no aliasing occuring as a side effect.