A builder's perspective on the
advantages:
tone - consistent from fretted notes to open string. Probably not noticeable to some, but for those who hear it and like it, it's immediately noticeable.
construction -
easier to get right than making a nut. I just cut an additional slot and install one more fret. There's a touch more to it, but not much. The 'nut' then becomes a string spacer and as a bonus, you can make it out of whatever you want. I usually make them out of fretboard material for a cleaner look.
Disadvantages:
People tend to associate them with cheaper quality instruments. They used them
because you don't have a have a well-cut nut to have a well-intonated instrument with a zero fret, so those manufacturers saved some time and money and increased consistency in their instruments by using it.
There aren't really any other disadvantages to the player by having a zero fret.
Some of my zero frets:
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