Duct tape. Many more things you can do with it that you cannot do with string. I have a roll in my tool kit, one in my BOB, another in the car, and well, even a few odd rolls hanging out on book cases in my LR.
I even saw colored duct tape used as an upholstery patch at a restaurant yesterday morning.
Both. Even my smallest kit has 50' of nylon cord, 6' of duct tape, and 3' of wire. Cordage is one of the hardest things to manufacture in the field but one of the most useful when it comes to making shelter. That said, when your down jacket tears, it's worthless. I wrap duct tape around everything so it's always available. I have 3-4' on each ski pole, I have 10' wrapped on my avalanche shovel, I have 3-4' on my seat post, I have 2-3' wrapped around my pocket first aid kit... it's everywhere, but try building a shelter out of four feet of the stuff and you'll be hard pressed.
I'd have to vote for Cordage (string is weak) in a 'SURVIVAL' situation, that's why I carry quite a bit of it when I'm out hunting or whatever...
In the general Home/Farm use, I'd have to say Duct tape.
Works good for a lot of things around the house, but has limited uses in a 'SURVIVAL' situation.
A third option, and the secret us farm boys keep from all you 'City' people is 'Bailing Wire'.
Works great for a BUNCH of things, and I once had a '63 GMC pickup that was held together entirely by bailing wire and concert/bumper stickers!