Simply put, scanning film often gives better results than scanning prints. One obvious reason is because the film is the original image instead of a second generation copy. This means that film contains much greater detail than is possible in prints. Film also has much greater dynamic range (contrast) than prints. Prints have already been manipulated, some tonal range lost, some color data has been modified, the total area has been cropped, etc, and we cannot get that data back from prints. These differences are very real, and critical commercial work normally scans film, usually slides.
I certainly don't mean to imply scanning prints is unsatisfactory, because we all obviously get good results scanning prints. The one main case in which you are likely to be unhappy when scanning color prints is when you enlarge the size, for example, scanning a small photo to print it double or triple size. For reasons of resolution and detail just explained, enlarging larger images from smaller prints is unsatisfactory if you go very far. Prints don't have enough detail to give for this purpose. Print paper and film are designed very differently for different purposes, and enlarging from print paper was never one of these considerations.
The photofinisher likely could not retain detail in the highlights and shadows in the photo print, and so our scanning those prints will never get that back. But scanning that print should be able to match what we see now in the print, but which is less than is in the film.
The basic scanning procedures, the histogram to set the Black and White Points, adjusting image brightness with the Mid Point, Unsharp Mask sharpening, scaling, printing, etc, it's all the same concept for film or prints, no difference.
Scanning film is the same procedure as scanning prints, and I think this entire material is about scanning film too. But there are of course a few differences, and here are a few observations (with some repetition).
The prints probably have already been cropped, but the film still has 100% of the frame.
The resolution and scale factor numbers are typically larger when using film, because film is typically small. The arithmetic to scale the size to be printed on paper is the same procedure, but the film is scanned at numbers around 2700 dpi instead of 300 dpi. The Scaling factor may be 1000% or more. This compensates for the film area being much smaller than photo prints.
Scan one inch of film at 1800 dpi and get an image size of (1x1800) = 1800 pixels.
Scan six inches of print at 300 dpi and get an image size of (6x300) = 1800 pixels.
Print either 1800 pixel image scaled to 200 dpi and get (1800 / 200) = 9 inches on paper.
The big difference is that film can easily produce real detail at resolutions like 2700 dpi, but the color photo print simply is not going to show much additional detail beyond about 300 dpi, if that (Chapter 8). The print is typically larger than the film, so the results (size and detail) when scanning 4 inch prints might compare to scanning 35 mm film at 1200 dpi.
A result of this is that since real detail is easily available from film, you are likely to be working with images that are 10 to 60 MB in memory size, maybe more. You will want 128 MB of system memory for film, and more would be better. 256 MB is nice for large images, and even that would be marginal for a 16 bit 4000 dpi image (120 MB if 35 mm). For example, multiple layers in an editing program would multiply that size for each image layer. History Undo information also consumes memory.
Conversely, images created by scanning photo prints will not likely approach 10 megabytes (little point in it unless scanning larger prints, like 8x10). So, more memory is needed for scanning film, simply because film can do it.
The second major benefit is that film can offer much greater contrast range than can prints. Slides can have extreme density range and frankly, the best scanner hardware is none too good to go deep into the dark areas of slides. Inexpensive scanners have relatively poor dynamic range, which cannot go deep into the detail in the dark shadow areas of slides. Even worse, inexpensive scanners also have relatively high noise levels, so that those dark tones contain random electronic scanner noise. The dark areas are black, near zero signal value, where the low level noise is. The dark signal is down in the noise level of the scanner. Slide film extends the black tones nearly two orders of magnitude deeper into the zero pit than do prints. Reflective prints don't have nearly the contrast or range as film, and inexpensive scanners can give good results with prints.
Repeating, low dynamic range means the darkest areas of images scanned from slides are black, and without shadow detail. The scanner's range could not reach that far down. Those black areas are also specked with random noise pixels, with an unattractive speckled appearance. It can be pretty bad in $200 scanners. Top end scanners are much better, but are still far from perfect in this respect. Scanner noise is a continual concern when scanning slides. Such scanner noise is seen as many random colored pixels in dark areas of slides, like continuous speckles.
Multiscanning is a technique to overlay and combine several different scans of the same film image, for the purpose of averaging out this random noise. This helps slide images to show dark areas cleanly (still without deep shadow detail, but the noise won't hide detail). Even top end film scanners benefit from multiscanning slides. Some support multiscan, and VueScan supports even more models.
Negative color film has much less density range than slides (slides can be near 3.6, negatives about 3.0, prints about 2.0, Chapter 19) and therefore inexpensive or 30 bit scanners are often more acceptable for negatives than for slides. Negative images are inverted, dark becomes light, so that any scanner dark noise appears in the highlights of the image, where it is usually much less detectable. Noise is seen in light areas of images from negatives (like skies), and in the dark areas of images from slides.