Which Way is Better Between DI Box and Microphone

Nobody can.

These two techniques are just different:

Direct Coupling (or Insert) is using the electrical signals, the way they are generated by the instrument. Of course we are talking about electric or electronic instruments. Like e.g.electric guitar.

This results in a pretty clinical, poor sound, and misses most of the personal touch of the performer. (see further)

Acoustical coupling:

Sound transducers and/or microphones.

Musicians chose their instruments and want to create "a specific sound". They experiment with different amplifiers, effect boxes, speakers. Some like to partially work in the cut-off preset of the preamp to create distortion, over overpower the speakers that "clip" the sound. The effects are very difficult, if not impossible to re-produce in sound recording studios.

Position to- and distance between the speaker, and the microphone add to these gigs and also the acoustics come into play. (room effects like echo and reverberation time and amplitude

*-* check the difference between life and studio performance records)

Also the type of microphone is pretty important. It changes the "timbre" too.

Base transducers need to work well in that range, cymbals need high pitch - and so on...

It is not question of better:

It is just your job as a producer- sound engineer to record a by the artist accepted sound. This is more like choosing colors - a feeling - thing.

It is a abstract communication- where sometimes terms as "more brown"- pointer sisters punch- and other codes are exchanged.

This also counts for Public Address systems for Life performances. Sometimes both are used together and mixed on separate inputs for e.g. more wedge monitor distinction.

If you want more about this, you can email me. I have a lot of studio and stage experience. Worked with names.

I still have room for 2 answers:

One:

That performer is probably using a acoustical electric guitar - one that has a hollow sound chamber. If he is e.g knocking on it to produce a drum like sound - a DI box will not pick it up that way and you will hear only some string vibrations. (Gibson?) Two:

The input impedance of your DI box can have a mismatch towards his pick up element (or incorporated pre- amp as some guitars have). This can make the sound "lack" harmonics and even create a mis-balance (I don't use unbalance here) resulting in poor sound.

a too low input imp. steals your highs. Not three:

If the guy plays on his own amp and speaker, try to copy the sound there, unless he is not happy with his own sound out of his box, or he wants you to perform tricks out of your side effects collection. If you go for a microphone, do a life check first. Don't take one with a too big voice coil and very directional. Vocalist with guitar is not easy - they forgot an orchestra to balance it out. Couple of beers makes decisions easier.