What Happens if you Leave New Potatoes Growing?

This is probably a really daft question!

Due to dying plants I have started to lift my vivaldi potatoes earlier than I planned, I was hoping to leave them in to keep for storage.  I have started using the small vivaldi's as new potatoes (very nice  ).

I also planted some vanessa early potatoes and they are currently flowering and looking really healthy.  If I just leave these to grow will there be any effect on the flavour or texture of the earlies?  Will they just get bigger? Can you store early potatoes over winter?

Questions questions!

Thanks in advance.