>Similar to using noise shaped dithering to allow you to hear more than the theoretical dynamic range in audio
I’m not sure how apt this analogy is. Dither is only useful for mitigating artifacts of quantization noise; it does not generally increase perceived detail of any arbitrary signal. More specifically, it is introduced to reduce the harmonic content of quantization noise, at the expense of a higher overall noise floor. Suppose you have a 1kHz sine wave; quantizing it will introduce harmonics that peak at, say, an average of -70dB, with an absolute noise floor of -120dB. Adding dither will raise the noise floor to -90dB, but reduce the harmonic peaks to -100dB. So while it actually decreases the true dynamic range, it increases perceived signal quality by removing the harmonic content.
These spurious harmonics occur because quantizing a signal introduces periodic artifacts. For example, suppose our analog sine wave can continuously vary between 0-7, and we quantize it to 3 bits (discrete values 0,1,2,…,7). Any analog value 4.7 will always be rounded up to 5; in a sine wave, the value 4.7 will occur periodically, thus resulting in a periodic rounding artifact, leading to harmonic distortion.
In order to prevent these periodic rounding errors, dither needs to be added pre-quantization, so that 4.7 can sometimes randomly become 4.4 and get rounded down to 4 during quantization. Adding “dither” to an already quantized signal (e.g. digital video) would just make the apparent picture noisier.
As a test, try quantizing a full-color image to 16 colors, and then adding back some random noise. It won’t look any better. You need to strategically dither the 16 colors with knowledge of the original full-color image.