Does the freshness of a potato matter? Specifically, does a "just-harvested" potato have any better qualities than say a properly stored one-month old potato?

Potatoes are the ideal vegetable, for both freshly-harvested (aka
'New') and for long-term storage. The difference between a new and
storage potato is in the water content. As time goes on a potato will,
under ideal storage conditions, lose internal water percentage. Other
than that there are few, if any differences, in performance and flavor.
Here's a link to our potato storage infographic:

https://idahopotato.comhttps://cdn.idahopotato.com/uploads/media/Infographic-Potato-Storage-Tips.pdf