Question: With the legalization of moonshine in many US states I've been wondering what are the big differences between Moonshine and Vodka?

Both are clear spirits made from whatever grains/starches are available. Both can be fairly high proof.

Is it all just an Appalachian that is tied to their origin?

Answer: 
If you want to be a traditionalist, it's not really Moonshine unless it was produced illegally, without all the proper taxes and such. Anything you buy in the store could more accurately be called white liquor, grain liquor or corn liquor, depending on the mash stock.

A key feature of both legally and illegally produced Moonshine is that it's unaged. It goes straight from the still into the jar or jug. This facet was born of the need to make quick profits, the need to minimize the amount of evidence on hand in the event of a raid, and by extension, get your liquor into the hands of paying customers before the revenuers could find it.

Physically speaking, there is no real difference between vodka and moonshine. Both are unaged neutral spirits, usually cut with water to increase volume and produce a more drinkable product. The difference is mostly geographic. Moonshine, at least commercial varieties, can easily be called American Vodka, much like Poitin could be called Irish vodka.
As a detail and hopefully and appropriately informative aside for this topic:

In chemistry lab there are two alcohol solvents used:
1. Medical Grade (surgical alcohol) = 95% Alcohol (getting that last 0.57% is very time consuming to achieve)
2. (benzene) Alcohol = 98% Alcohol (with 2% benzene still in solution with the alcohol)

[Benzene is used to drive off or get all of the 5% of water out of distilled alcohol](https://www.google.com/search?q=Benzene%20is%20used%20to%20drive%20off%20or%20get%20all%20of%20the%205%25%20of%20water%20out%20of%20distilled%20alcohol&oq=Benzene%20is%20used%20to%20drive%20off%20or%20get%20all%20of%20the%205%25%20of%20water%20out%20of%20distilled%20alcohol). The (benzene) Alcohol will make you sick because of the benzene. 

Surgical alcohol provides the most wonderful drunk. It is a body drunk with zero dizziness. 

Dizziness comes from the long chain fusel oil(s) (alcohols) in fermented alcohol. Fusel oil alcohols are not present in surgical alcohol. All fusel oil alcohols boil off at a different, slightly higher temperature than ethanol (drinking alcohol). Sloppy or rushed distilling gets the higher temperature fusel oil alcohols coming over with the ethanol. Really sloppy and very rushed distilling gets the first run methyl alcohol, methanol . . . which causes permanent eyesight loss. Long chain alcohols cause much of the hangover effect of headaches.

Yeasts produce fusel oils when they struggle to produce higher percentage alcohol levels in wines and beers.
[https://www.google.com/search?q=FUSEL+OIL+ALCOHOLS](https://www.google.com/search?q=FUSEL%20OIL%20ALCOHOLS)