On our trip through Italy, we toured through many churches and museums. In some places, photography was forbidden (and they mean it! Learn how to turn your flash on and off ahead of time.). In some areas photography was okay, but no flash. And many allowed flash photography.
Sometimes I would take a flash photo, and if I wasn’t sure it turned out well, I would repeat the photo with no flash.
In St. Peter’s Basilica in Vatican City, photography is allowed because there are no paintings. There appear to be paintings, but they are actually mosaics. I thought this photo of the interior of the dome of St. Peter’s was an interesting comparison. First, with flash:
Second time, without flash:
Both photos are okay, but I prefer the second one with no flash. It looks richer and a little brighter to me. Sometimes, though, I think flash works out better, and sometimes no flash results in an overly golden look (which can be pretty or you can correct it with editing).
Another consideration is due to the length of exposure time needed in a low-light situation without flash, you need to hold very still. If possible, steady yourself against a wall or something solid. (For comparison, the exposure time with flash was 1/30th of a second; without flash was 1/10th of a second.)