The potato will grow till the vine dies so dig them when they are the proper table size. But they must be dug before a frost or the soil temperature reaches 50 deg. They are very sensitive to cold temperatures and just a few hrs of cold temperatures at any time can cause them to rot. Get them out of the sun as soon as possible and keep bruising to a minimum.
They will not be sweet when dug. They need to be cured at 80 to 85 deg and 85 to 90 percent humidity for at least 10 days. If you can not get this storage cure them longer such as 65 to 75 deg for 3 weeks.
Once cured store them above 55 deg.
So what this means to a home owner is bring them inside as soon as you dig them; wash and put on counter. Do not ever put in fridge. Do not eat for over 3 weeks to give them time to sweeten.