An ADC merely converts an analogue signal to a digital one. Why then do we have special ADCs for audio and video application? What if one was to use general purpose op-amp in them instead?
First, I think you use the word "merely" very lightly. ADCs are some of the most complex, challenging mixed-signal systems in use. To meet different performance targets practical ADCs use many different architectures (the most important right now are sigma-delta, SAR, and Pipelined).
ADC design is characterized by very painful tradeoffs between speed, accuracy, noise, and power dissipation. For example, to increase the SNDR of a thermal-noise limited ADC by one bit increases the power roughly 4X (because noise is proportional to sqrt(C)).
To first order (and this is VERY rough, mind) you can think of the speed-accuracy product of an ADC to be constant. So, to make an ADC very accurate it must be slow, and, conversely, the fastest ADCs (now at 40 GS/s and beyond) have very low resolutions (4 - 6 bits or so). This is due to a combination of factors such as oversampling (taking multiple samples and averaging) which reduces the speed, and the capabilities of sample-and-hold circuits to acquire signals at needed accuracy (for example, a 10-bit sample-and-hold needs to sample the input signal to an accuracy of about 0.1%. A 16-bit sample-and-hold needs to sample the input to an accuracy of about 0.001% (!). Accuracy, namely signal settling, takes time.
So, an Audio ADC is typically 16-24 bits, and has an effective sampling rate of 44kHz to 96 kHz or more. (keep in mind the ADC is sampling MUCH faster than this because of sigma-delta modulation).
A video ADC is typically 8-12 bits (sometimes 14b) and samples between 10 - 40 MHz.
An ADC in a Gb Ethernet chip would be more like 6 - 8 bits @ 125 MHz.
An ADC in a 10Gb Ethernet chip would be more like 6 - 8 bits @ 1.25 GHz.
An ADC for a DDR4 transceiver or a radar receiver may be more like 4 bits at 10 GHz.
And so on. The reason there are so many ADCs is that there are so many places in the parameter space. Do you care about noise? It will cost you. Do you care about power? It will cost you.
A general-purpose ADC is a balance of different factors that has use in a variety of applications, but can't perform at extremes of speed or accuracy.
A general-purpose ADC is expected to work reasonably well capturing any kind of frequency content from DC up to its bandwidth limit; the accuracy at recording any particular frequency content should be unaffected by the presence of other frequency content.
An audio ADC is expected to work reasonably well at capturing audio signals in a range which does not extend to DC and typically doesn't go as high as many general-purpose ADC devices. While the accuracy with which a general-purpose ADC can capture content at one frequency should be unaffected by other frequencies, that is not so much of a requirement with audio ones. The ratio between the loudest allowable signal and the amount of noise on the quietest detectable signal should be high, but it is often acceptable for the ratio between the loudest signal and the amount of noise that would be present on that signal to be much lower.
A video ADC will need to operate much faster than many general-purpose or audio devices, but will often not require particularly good specifications for linearity. Further, while the delay between a signal appearing on the input of a general-purpose ADC and its being propagated through to the output will usually be no more than two sample times, that time may be longer on a video ADC. When an ADC is used in a control loop, it's important to minimize the the delay through the ADC, but when delay is acceptable it allows the hardware of the ADC to be simplified. An 6-bit ADC that needs to yield "instant" results will need 63 comparators, and an 8-bit one will need 255. Such a design is called a "flash" ADC, and the cost of flash ADCs increases exponentially with resolution. If one doesn't need an ADC to be fast, once can use a successive-approximation ADC which takes a certain amount of time per bit, but only requires circuitry proportional to the number of bits.
A modern video ADC will work as a cross between the two approaches. Any particular sample will have to be processed in multiple steps through different parts of the ADC, but different parts of the ADC can all work simultaneously. Once the first stage of a pipelined ADC hands a sample off to the second stage, it can begin work on the next sample immediately; it doesn't have to wait for the remaining stages to finish their work on the first sample. A video ADC might this take 125ns to process a sample and yet be able to process 64 million samples per second. The fact that data from the first sample won't be available until after eight more have arrived is a tiny price to pay for having number of simple stages rather than one really massive one.
General Purpose ADC - These are the general run of the mill ADCs that you would use for your load cell, temperature sensor, whatever. They come single ended and differential, all sorts of
Audio ADC - There are two things that can make audio ADCs special. First, they usually have AC coupled inputs, meaning instead of a 0-5V range, they'll have a -2.5-2.5V range. Most these days will be 24-bit. More often than not, they'll have an I2S interface, and an onboard DAC for each ADC.
Video ADC - Consider old school 640x480 VGA signals. It had a 25MHz pixel clock. Video ADCs have to be insanely fast.
In broad-brush terms, it's largely about bandwidth, and quantisation fidelity.
Audio requires relatively low speed conversion rates (e.g. 44.1kHz), but typically 16bit resolution or higher conversions.
Video is far higher conversion rates (several MHz), but usually much lower resolution - say, 8, 10 or 12 bits per colour channel (RGB).
There are other applications where an "audio" or "video" ADC is appropriate, too.
And it's important for these types of applications (and others) for the ADCs to sample multiple channels simultaneously (e.g. left + right audio, or R G & B video), which essentially doubles/triples the number of ADCs needed in the system/on the chip (which is of course more expensive), whereas "general purpose" ADCs are often NOT able to sample multiple channels simultaneously, they're just a single ADC with a multiplexing switch in front of them to select the desired channel - so if you want to sample all 4 channels, you sample #1, then #2, then #3, then #4. If that difference in timing between when the samples are taken doesn't matter to you, then this kind of multiple-input ADC is fine.
As for the op-amps involved in the front end analog circuitry of ADCs, again these are specified for a bunch of parameters that are appropriate to the bandwidth & amplitude of the signals being manipulated - GainBandwidth-product, input offset voltage, voltage ranges, linearity, etc. The term "general purpose" op-amp might be a bit mid-leading, but it basically means it doesn't excel in any particular spec; whether or not that's good enough for a particular application is up to the designer to calculate.
edit: also, many ADCs have differential inputs, which usually require differential op-amps.