For example, when I watch Eric Clapton live, playing his acoustic guitar, he uses jack output and a mic.
Why is it? Is it for acoustic nuance?
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As others have noted, the properties of the signal from the microphone and from the piezo pickup will be different. The microphone picks up the same kind of air vibrations your ear does. The piezo pickup picks up the vibration of the saddle.
The pickup has the advantage of being less susceptible to (but not immune to) feedback, and it moves with the guitar.
The microphone has the advantage of getting a more natural sound, but several practical disadvantages such as:
If, as a sound engineer, you have both signals, you give yourself a range of options:
The sound from a microphone inside the guitar (or piezo pickups as in the case with Clapton) is different from the sound outside of the guitar. The signal from internal microphones or pickups will be more consistent, since it is not affected by movements of the guitar. Likely the signals are mixed to get the best of both worlds.
Adding to Meaningful Username's answer, the different sources (piezo pickup vs microphone) emphasise different parts of the guitar's sound. Not only is it used to help the sound be more consistent etc. but the different sounds themselves make for much more versatility when mixing.
In general (this can vary wildly) a microphone will give a more "natural" sound (ie. closer to what we perceive when listening to it un-amplified) and can be described as "full sounding". Of course this depends on the microphone and placement. The pickup will often give a much more "up front" sound, emphasising the mid range more than our ears do. By itself sometimes it can sound a little thin but can really make the sound stand out when mixed with the microphone source.
As mentioned, this is definitely prone to variations. Some pickups have enormous frequency range and some microphones are very thin or up-front sounding.
The sound from the microphone is undoubtedly the best, if you want the music to sound as close as you can to how it sounds unamplified. The mic captures all of the natural resonances of the guitar without the "quack" of a piezo pickup.
However, controlling feedback with only a microphone is difficult. The musician will have to keep their guitar relatively still while playing. Blending the microphone sound with sound from a piezo pickup helps to mitigate these issues.
Some artists (for example, Gillian Welch and David Rawlings) exclusively use microphones for their guitars. However as you add more band members, this becomes more and more difficult to set up! This excerpt here is from an interview in the April 1999 issue of the Acoustic Guitar Magazine.
Unless the artist is fanatically dedicated to the sound quality of their performance, they'll add a pickup to make set-up easier.