I guess it's good that we have choices. Personally, I think an octagon barrel is about the ultimate in elegant modifications that can be done to a revolver.
On weight, just as with round barrels, it's all about the contour. With single action revolvers, you're restricted by the platform as to how heavy or light an octagon barrel can be. The window is very narrow. For this reason, they are only slightly heavier than their round brethren. On rifles, it's a different matter as octagon barrels tend to be a significantly heavier contour than their round counterparts, on purpose. Particularly on leverguns, where carbines are meant to be light and handy, they have round barrels where "rifle" configurations often had octagons. Same for single shots. In muzzleloaders, an octagon barrel can range from the heavy Hawken type, meant to withstand heavy loads suitable for buffalo or the svelte, swamped long rifle (Pennsylvania/Kentucky) which may have a 42" or longer barrel but an overall weight of 7lbs. Meanwhile, the very slight increase in diameter between my .54cal Lyman Great Plains Rifle and .54cal Pedersoli Rocky Mountain Hawken adds up to a full pound.
How much that weight matters is completely subjective. Some folks balk at it, I don't have an issue hunting with a 9-10lb octagon barreled Hawken or a 8-9lb octagon barreled levergun.
The difference between this custom .54cal flintlock pistol by Steve Zihn (29oz) and my Lyman .54cal Plains Pistol (41oz) is not due to the shape of the barrels but to the much heavier contour of the Lyman.