The first thing to remember is that bit depth and dynamic range are NOT the same thing. It is going to sound much the same, but it's not. That difference will be covered here.
Scanners today, and digital cameras too, are 12, 14, or 16 bits in each RGB channel, which ×3 is 36 or 42 or 48 bits color depth (with data values 0..4095, or 0..16383, or 0..65535). Specifically, more dynamic range may allow more detail in the shadow tones of images from positive film (slides), and in the highlights of images from negative film. Dynamic range is not a major consideration for scanning photo prints, because the dynamic range of the paper prints themselves is very limited, so this is far more important when scanning film. We'll try to explain HOW it helps.
More bits support more dynamic range, but does not ensure it exists in the scanned media. More bits are required to hold numeric values containing better dynamic range, but this one detail does not ensure the data exists or is actually measured. There is also a second factor. High-quality low-noise CCD and electronics are needed for better dynamic range. The trend today is that inexpensive scanners are offering 48 bit A/D conversion (analog to digital), which just means that inexpensive A/D chips are available now. Don't assume that a $100 48 bit scanner can offer as much as a $1000 36 bit scanner can, but the 36 bits is likely sufficient for any media.
Image density is measured from image brightness with an optical densimeter, and ranges from 0 to 4, where density 0 is purest bright white and 4 is extremely black. More density is less brightness. Density is measured on a logarithmic scale (similar to wind speeds or earthquakes). Density of 3.0 is 10 times greater range than a density of 2.0. The "range" of density is the ratio of the maximum data value / minimum data value, which is called Dmax / Dmin. An intensity range of 100:1 is a density range of 2.0, and 1000:1 is a range of 3.0, and 10000:1 is a range of 4.0. Because, Log10(100/1) = 2, Log10(1000/1) = 3, Log10(10000/1) = 4. The 16 bits can theoretically be values 0..65535, and Log10(65535/1) = 4.8 (theoretically). Meaning the maximum theoretical value that 16 bits can store (if it should actually exist), but Dmax means the value must be distinguishable as unique from similar values.
The math is not otherwise limited, but no media that you can scan will reach 4.0. Color photo prints on paper won't exceed 2.0. And if you have blank areas at either end of your histogram, that is dynamic range that you are missing. 14 bits can store values of 0..16383, so it could technically store density 4.2 if present and if Dmin was not too high. But very many digital cameras only have 12 bits, which frankly seems very acceptable (absolute theoretical maximum is 3.6, ignoring Dmin). And the range of any 8-bit JPG file is 0..255, or 2.4 absolute theoretical maximum.
The minimum and maximum values of density capable of being captured by a specific scanner are called DMin and DMax. Dmax is the highest density value where detail can be distinguished, and Dmin is the lowest value distinguished. DMin implies the lowest density values that are not hidden by noise. If the scanner's DMin were 0.3 and DMax were 3.1, its Dynamic Range would be 2.8 (subtracted difference). Greater dynamic range can detect greater image detail in dark shadow areas of the photographic image, because the range is extended at the black end (into the noise).
Slides have more dynamic range but less photographic range than negatives. Slides have perhaps about 5 or 6 f/stops of total scenic range, compared to perhaps 9 f-stops for negatives. Expose a slide half a stop off and the results are objectionable. Do that with negatives, and you may never even realize it (then the printing process can improve exposure, but the paper media decreases dynamic range).
Note that Film Gamma (contrast curve of film density, next below) and Gamma Correction (to correct CRT monitor response, lower below) are very different concepts. Gamma is just a Greek letter, a very convenient label to use in many math equations (much like X in algebra).
Side film itself has more contrast, a steeper film gamma curve, so the larger density range is in fact less f/stop range. But while the captured scenic tonal range may be less, the density extremes on the film can be greater. Meaning, the extremes on slides are more likely to be clear or black film, but contain greater dynamic range as seen at the scanner. This is NOT speaking of digital camera images, which are positives like slides, with similar exposure concerns (Negatives: expose for the shadows, more can be better. Positives, expose for the highlights, don't clip them). But cameras have their own adjustable contrast curves. Color negative film has the orange mask (helps printing reversal color balance) which considerably increases DMin which decreases the overall density range. Images from negatives invert dark noise to be in the highlights, less noticeable there. Slides are much more difficult than negatives to capture the shadow detail, and slides do need a scanner with greater dynamic range.
Even 8-bit scanners might have a dynamic range specifications near 2.4, needed for photo prints. 10 bit scanners might be near 3.0, needed for negatives. The best 12 bit scanners might approach 3.6, better for slides. Only rotating drum scanners can actually approach 4.0 (these use Photo Multiplier Tubes, PMT, expensive). And all scanners are not equal, some will have higher dynamic range than others because their electronics have less noise. Price is definitely a factor.
The big problem with 8 or 10 bit systems is not so much dynamic range (except maybe for slides), but gamma correction conversion suffers. All tonal images (color, grayscale) are gamma encoded, decoded at display. 8-bit gamma 2.2 values of 0 to 14 all decode to 0 linear. 8-bit gamma values 15 to 24 all decode to linear 1. Gamma values 2 and 3 are not much different, and 8-bit gamma values do not have a single linear conversion value until gamma 21 (linear 82). See a gamma lookup table. This precision loss loses much of the possible dark detail. Gamma is one reason why all scanners and cameras today are 12 bits internally, but except for Raw images, they can output 8-bit JPG images.
Greater dynamic range extends the signal into low black levels where the noise is, which is the Dmin subtraction. To be effective, the electronics are improved to reduce the noise. The hardest problem for the scanner is the black end, density values beyond 3.0. One good reason is low level signal and noise.
Caution again: It is easy at first to assume that more bits will yield data with more dynamic range. Superficial reading might even think I said that in places. But it is NOT true at all. More bits are simply the container needed to store the data, but the bits do not create dynamic range, they only allow it to be stored. In the same way, perhaps a large wallet is needed to hold great sums of money, but having a large wallet does not necessarily imply the money is present. Many wallets are not exactly full.
24 bit color is three 8-bit bytes, one byte for each of the Red, Green, and Blue CCD channels, to describe the color of each pixel in the image. 30 bit color uses 10 bits for each of the three primary RGB colors. In binary numbers, each bit is a power of 2, meaning that each additional bit doubles the maximum size of the numbers that can be stored. Same concept in decimal numbers using powers of 10, each digit there allows numbers 10 times larger to be represented.
Scanners detect light intensity corresponding to the density of the original. More film density lets less light come through, or more print density reflects less light. The CCD sensor measures that resulting light intensity. The image RGB numbers stored are proportional to intensity values in the original.
Basically the human eye responds to brightness in a logarithmic manner. The human eye does not perceive twice the intensity as being twice as bright. For a common example, photographers use their light meters on a "standard gray card" made to reflect 18% of the light falling on it. Metering from that card is used to calibrate middle gray (50% to our eye) in the hypothetical "average" scene. We see that 18% intensity as apparent 50% brightness.
The 12 bit scanner divides the scanned density range into smaller steps, 4096 steps in 12 bits instead of 256 steps in 8 bits, and therefore can show slightly more unique detail in the shadow areas, for a couple of reasons. Tiny variations that might be the same one color value at 8 bits could be 4 slightly different shades at 10 bits, or 16 slightly different shades at 12 bits, or 64 slightly different shades at 14 bits. Tiny differences, and it is really only significant at the black end, but that's more detail. And the possibility of larger numbers provides an opportunity for a better CCD to extend the dynamic range a little way into the next "10 times" logarithmic density interval. The better CCD is required to capture this detail in the darkest film, and more bits are required to store the numbers representing more range..