It looks different. More of the grain is apparent with the dye since you are looking through clear tot he stained wood rather than looking through tinted paint to natural wood.
Dyeing black also doesn't take as much of the color of the wood to the final color. With the almost black tinted paint, there is always a slight greenish cast to the finished look since the wood is a little on the yellow side.
Both finishes look nice, but they do look different.
