I'm not sure about other players' reasons for it, but I tune first with pegs and then fine tuners because it's easiest to get the general pitch and then adjust with the fine tuners to the right pitch. Especially if the violin is really out of tune.
As the last person said its to get the most accurate intonation. Though you may have also noticed violinists who have violins with only one fine tuner on the E string.
As strange as it may sound (couldn't avoid that pun) a violin generally sounds better without the fine tuners. Advanced players use only the pegs and the one fine tuner on the E string, if they choose to have one.
Good violins have good pegs and peg-holes that easy to tune the violin up
without fine turners. You may try "peg drops" to see how it work on your
Fine turners are metal, that badly effect the beauty of the tone.
To improve the tone quality, please, eliminate all metallic parts on your
violin, including strings.
If you have any of your accessoris that are metal, please, turn it tight.
Don't leave any parts of it loose.
However, to common violins (not for famous soloists), metal parts do not
make any difference. Please, feel free to have your 4 fine turners.
If you were to only tune with the fine tuners ,you would end up with the fine tuners scrude in all the way. It is best to keep them about 1/2 way scrued in, then when you use the pegs to get it close( (at least on the right note), you can then get it the rest of the way with the fine tuners. Up or down what ever way you need to go. Also the best thing if you want the sound to be best and still like 4 fine tuners,is to get the tailpiece with the fine tuners built in.
Why do the better violins sound best without the fine tuners?
Think about this. It is not that they are metal! When you add a fine tuner to the tailpiece you shorten the length of the string from the back of the bridge to the tailpiece.
It is this shortend string length that makes the diffence in the sound. It mostly is notest in the G,and D strings . The E string is the least effected by this shorter dissence.
What it does to the sound is that it makes better carrying power and a little more sizzle when the length is right. on lower Quality violins it could make it sound to wild . In that case try a longer tailpiece.
The genral length from the tailpiece saddle to the back of the bridge is about 55mm.
All this is why I said in the last post that if you get a tailpiese with the 4 fine tunners built-in,it will not effect the sound . The string length would not be shortend.