I'm aware of two main differences between noise and grain.
First: noise happens to individual pixels, while grain crosses pixel boundaries. In colour film "grains" are actually dye clouds which are much larger than the individual silver grains seen in b&w. But even for b&w film the grain does not line up such that a single pixel is either a single element of granularity or a single element on subject matter. In practical terms digital noise is more a pinpoint effect while granularity is more spread out.
Second: noise in Bayer pattern systems has a colour component as well as a luminance component. Noise Reduction software can be set to remove colour noise, while leaving luminance noise. This not only comes closer to resembling film grain; but also chroma NR involves much less detail smearing than luminance NR.
The upshot of all this is that digital noise tends to look very different from film grain. To my eyes at least that look is less an organic part of the image and more an arbitrary intrusion to the image. More like measles and less like freckles. ;) But that's an aesthetic judgment and it may be that a generation that grows up with digital noise instead of film grain may have just the opposite response.