Yes and no... My wife uses her smart phone for grocery shopping because she has an app that keeps a list of the stuff she commonly buys. Throughout the week as she thinks of something she needs she adds it to the list if it wasn't on there already or simply selects it from the list of previous items. Before she goes to the store she might review the list for the next trip, and look at the list of previous items to see if anything sticks out as something she needs again and add it. Then when she's at the store she has the list and just marks the items off as she goes.
That implementation is better than just writing on a scrap of paper because it's something she is updating constantly. You could do that with the paper but then you need to keep track of one more physical item through out the week. Additionally since it keeps a historical record she can see what she bought previously, which helps to determine if it's time to buy more, and reduces the data entry requirements as it can just be added to the current list instead of typing it in again.
It's a clever app and I should probably use it myself. But I honestly hate carrying a cell phone so I just scribble down whatever she tells me to get before I leave.