I don't carry either/or... it's always both or none. I have various combinations for various means of transport and portage difficulty. When backpacking, I seldom carry either, as I'll generally be WAY deep, and then it's easy to find good firewood that doesn't need cutting and splitting, just hand processing with a small knife to split if you absolutely have to. I also have a few favored sites where I've stashed a small folding saw/hatchet combo most of the way in, so I don't have to carry them in/out every time.
For saws, I have a full size 32" bow saw, 24" folding bucksaw, Agawa Canyon Boreal 21, Sven 15, and Silky Gomboy and Pocketboy (~2lbs down to 8oz or thereabouts.)
For hatchets/hawks/axes, I have about 9, I think, and they go from 5lbs down to 12oz.