There isn't actually any reputable evidence that Nero fiddled while Rome burned. In fact, it would have been impossible given the viol class of instrument wouldn't exist for a thousand years after Nero's time, and the fiddle wouldn't be invented in its specific earliest form for fourteen hundred years. Nero was an exceptional string musician, that much is pretty accepted, but he probably played the cithara (or kithara in its original Greek), an ancient form of the lyre an instrument similar to the lyre.

He just didn't play it while Rome burned.

Because while there isn't any reputable evidence Nero did anything musical while Rome burned, there are primary sources which indicate that once he received news of the fire while at his villa in Antium he made the 35 mile journey back to Rome with great haste to direct relief efforts. That is well recorded. There is one statement that Nero sang while Rome burned, made by Tacitus, but he spelled out very clearly in that account that there were no actual eyewitness reports of this. So it was just a rumor against primary sources placing Nero in charge of relief efforts.

With that said, Nero was a terrible emperor. To make my point on the sort of Emperor he was the evidence regarding the fire would amount to a few sheets of paper, but you would have a pretty hefty stack if you gathered evidence of his cruelty, which at times probably amounted to outright sadism. That's possibly what allowed the myth of his musicianship during the fire to spread and gain popularity. It was just modernized once the fiddle was invented.

Edit - Correction to the struck through portion above, because something didn't feel right about it the moment I posted it, and I don't want to give a misconception in a post about a misconception. The lyre and cithara actually coexisted, and the difference is in size and other details.