The fundamental difference between optical prints ("print from negatives") and digital prints "from scans" is that, with optical prints, as you increase magnification, you start to see film grain, but with digital, as you increase magnification (often not nearly as much) you start to see pixels.
Film grain in black and white is most visible in large areas of light to medium gray, where it will look much like fine dark sand on the white surface of the photo paper. Grain carries a signature of the film used to make the print -- a fine grain film like FP4+ or Fomapan 100 will have smaller grain than an old classic fast film like Tri-X -- and of the developer used to process the film that created the negative -- Rodinal or Dektol will produce crisp, unapologetic grain, while D-76 or Xtol stock (often used in commercial processing because they can be replenished) will smooth the grain through what's called "solvent action".
Color film also has grain, but it looks much different -- it's not produced by clumps of silver nanoparticles, but rather by clouds of colored dyes produced by the developing process (the silver is then bleached and fixed away to leave only the dye image).
A scanned image, however, is made up of a large collection of numbers, each representing the brightness (in a single brightness-only channel or in each of three color channels) of the light that was read from a specific, very tiny region of the negative. No smaller detail is recorded, and few if any scans can approach the level of fineness of actual film grain (this would be equivalent to at least 50 megapixels from a 35mm 24x36mm negative). Therefore, when you make a digital image larger, there comes a point where you start to see tiny squares, each a solid color or gray value. These are the pixels that form the image, and if they're small enough, they will appear to blend into a continuous range of tones -- but they will never show the grain of the film unless the scan has higher resolution than the film emulsion (which in only just possible with the best scanning technology).
In reference to "machine" prints, some of the earlier machine systems made optical prints as part of the processing, but these were typically small (three or four inches on the short side), small enough that there is no discernible difference in the end product between optical prints and laser prints from a scan. Later machines would scan at "standard resolution" (typically around 2 megapixel) in the main machine, but prints would be done from the scan files on a different machine -- and as such, might have been presented as an extra cost option, or the option to get negatives and CD only offered at a discount. Once again, the standard prints were small enough there's nothing to distinguish, by eye, optical from laser prints. Even in the later machines doing "in line" scanning, a higher resolution scan was generally done in a separate step on a different machine, and this would be desirable to maintain image quality if larger prints (above around 6x9 inches) were wanted.
One reason, other than the technology used, that your modern 35mm images might not have better quality than those your grandparents (or great-grandparents, these days) produced in their youth is that they most likely used LARGER FILM. A 120 negative, for example, will contain anywhere from three to six times the information in a 35mm frame, given identical film and lens quality. Now, films are better than than they were even in the 1980s -- but they're not enough better to make a new 35mm significantly better than a 1970 120 image. Further, back then, professional photographers often used large format (4x5 inches or even larger), which is another twice (or more) area of the largest 120 negatives.