The purpose of greater scanning resolution is to create more pixels, to create a larger image size (described in Chapters 4, 5, 6).
35 mm film is relatively tiny, requiring greater scanning resolution than photo prints to create an enlarged image for printing. The ratio of (scanning resolution/printing resolution) is the enlargement factor. For example, scanning film at 2700 dpi and printing scaled to 300 dpi gives 2700/300 = 9 times enlargement of the original film size. 9X is about 8x12 inches (near A4 size) from full frame 35 mm. This enlargement requirement is why film needs high resolution.
Most flatbed choices are 600 or 1200 dpi now, and some are 2400 dpi. You won't need more than 300 dpi for scanning photo prints, or 600 dpi for line art documents, assuming printing at original size. 1200 and 2400 dpi would be used for scanning film.
Flatbed scanner specifications are stated with two numbers, like 1200x2400 dpi. Flatbeds also usually specify a maximum resolution, like perhaps 9600 dpi. So what does all of this mean?
A scanner scans one horizontal row of pixels at a time, moving that scan line down the page with a carriage motor. The smaller dpi number is the optical resolution of the CCD sensor cells. A 1200 dpi scanner takes 1200 color samples per inch (creates 1200 pixels per inch) horizontally from the width being scanned. A 1200 dpi CCD sensor really cannot do anything else but scan at 1200 dpi. This rating does not mean that it can resolve 1200 lpi in a test target, but instead, the CCD simply reads 1200 samples per inch. Nyquist sampling theory says the image can never resolve more than 1/2 of that detail level, and in the real world, a little less.
The larger dpi number is the possible positioning of the carriage stepping motor. A stepping motor doesn't rotate continuously like regular motors. Instead it is pulsed to move in precise steps, rotating only a few degrees with each input power pulse. A 1200x2400 dpi scanner is geared so that each pulse of the carriage motor moves in 1/2400 inch steps vertically. If we scan at 300 dpi, the carriage moves eight motor steps at a time vertically, then stops and samples, and resamples the scan line to 1/4 size horizontally, to create the 300x300 dpi image requested. If scanning at say 250 dpi, it should move 2400/250 = 9.6 steps per row, but it can only move 10 steps on some rows, and 9 steps on others. Any location error will be less than half a CCD cell height, even in worst case. This is the purpose of the 2X rating of the motor. The purpose is NOT to scan at 2400 dpi. The motor does not contribute to optical resolution. A 1200x2400 dpi unit is a 1200 dpi scanner.
Most flatbed scanners also advertise a "maximum" resolution, 9600 dpi, or even more, but this is a meaningless number. It is simply interpolated resolution (see Chapter 13), and you can do the same thing blowing up the image later in a photo program (except you won't, the quality is blurred, not improved). Resolution greater than the CCD optical rating is simply interpolated resolution, done in software after the 600 or 1200 dpi optical scan. Interpolated resolution is the least important scanner specification. It was useful for line art mode, and only for line art, to reduce jaggies when we had 300 dpi scanners and needed 600 dpi line art.
The flatbed scanner bed is 8.5 inches wide, so a 1200 dpi CCD sensor is an array of (1200 dpi x 8.5 inches) = 10200 pixels in one horizontal line. A wide-angle optical lens focuses the 8.5 inch image width onto a much smaller CCD chip, using mirrors to fold the long optical path inside the scanner. A typical flatbed CCD array is perhaps 72 mm wide, with 7x7 micron cells (3628 per inch in this example) being popular today. The carriage motor moves the CCD scan line vertically down the length of the bed, taking a 10200x1 pixel scan line sampled periodically from the 8.5 inch image, at each image row location where the carriage motor stops. We call this 1200 dpi, and for all purposes it is, because at the glass bed, 10200 pixels / 8.5 inches = 1200 dpi.
A 35 mm film scanner uses a different optical lens which covers only the 0.9 inch film width instead of 8.5 inches. 4000 dpi over 0.9 inches is 3600 pixels, instead of 10200 pixels. That's a big deal, and the narrow width allows higher resolution, and in particular allows larger CCD cells, which means more CCD quality with less CCD noise. This larger sensor size is a big advantage for film dynamic range.
Digital photo images have "square" resolution, the same in both directions like 300x300 dpi, simply called 300 dpi. If we did try to scan film at 2400 dpi using a 1200x2400 dpi scanner, the carriage motor can indeed step at 2400 dpi vertically. However, all samples will overlap each other vertically by 50% because the 1200 dpi CCD cells are twice taller than 1/2400 inch in size. Horizontally, the CCD can only sample at 1200 dpi but our images must be square resolution, so the software interpolates larger horizontally to create a 2400x2400 dpi image. This will not be the same quality as a "true" 2400 dpi CCD can do, either horizontally or vertically.
2400 dpi flatbeds
Current consumer flatbeds over 1200 dpi build the CCD differently than others. These may not be exactly the same as "true" resolution either. A true 2400 dpi covering 8.5 inches (20,400 pixels) would necessarily require very small CCD cells (not so good). So flatbeds of this class use two staggered larger arrays (half-resolution 1200 dpi arrays) in one chip, called HyperCCD or MatrixCCD, if it is mentioned at all. Each array has three RGB sensors, so it may be described as "six rows" of RGB sensors. The two rows of larger pixels are staggered to overlap on half-cell spacing. Each scan line row of image pixels is scanned by both rows of CCD, and merged to create the stated 2X resolution number.
The overlap due to these larger cells blurs the finest detail at that tiny scale. Much more sharpening than normal is necessary if scanning at full 2400 dpi optical rating. A case can be made that there is possibly a theoretical advantage. Both the overlap, and also the higher sampling resolution it allows, reduce digital aliasing (false output generated from image detail that is too fine for the CCD to accurately reproduce). Certainly the overlap is no concern at normal lower flatbed resolutions like 300 dpi, since resampling blends several pixels together anyway. But if scanning at full optical rating, like say film at 2400 dpi, each pixel stands alone then, and the overlap does noticeably reduce sharpness. These are good well-rated flatbeds, and I am not saying otherwise. I am only pointing out that all current 2400 dpi flatbeds do this, and that it is a rather different construction than others. I'd suggest a real film scanner for serious film scanning.
CCD or CIS sensors
Most scanners conventionally use CCD sensors (Charge Coupled Device). These units use an optical lens, often like a fine camera lens, and a system of mirrors, to focus the image onto the CCD cells. The CCD is an analog device, also requiring an A/D converter chip (analog to digital). All this adds substantial expense and size, but most flatbeds do use a CCD sensor for best image quality (low noise, good dynamic range, and color uniformity).
The "compact" and "ultra thin" scanners use a very different CIS chip (Contact Image Sensor). These CIS units are small and inexpensive, having no optical system (no lens, mirrors, lamp, and no A/D chip). CIS chips often have LED light sources integrated in the chip with the sensor. The CIS sensors are full size, extending over the full bed width. They work by simply being extremely near the paper being scanned (as "in contact"). This means that there is zero depth of field above the scanner glass, anything not actually touching the glass is too distant to be sharp, making CIS unsuitable for scanning 3D objects. CIS is also used in sheet-feed scanners and fax machines where depth is not a factor.
Carl McMillan has posted a good example showing the absence of depth of field above the glass bed on CIS scanners.