The advantage in cutting down the haulm is to stop spores of Potato blight getting onto the tubers (mine went down early, and hard this year). If the leaves are blotchy remove the haulms, then wait for any spore on the soil to die before harvesting for storage (OK if you are eating now).
If you leave them in the ground you may lose them to slugs and rodents.
RHS says "With maincrops for storage wait until the foliage turns yellow, then cut it and remove it. Leave for 10 days before harvesting the tubers, leaving them to dry for a few hours before storing."
Let the skins of the spuds dry before you put them in store (I spread them on a paper sack, rub any damp soil off roll over and repeat). Store the perfect ones in a paper sack or cardboard boxes. And that are damaged or look odd use soonest. I keep mine till Christmas (not enough to last longer).