Amazon vs Netflix, closely resembles which type of competition? So we all know Amazon is following the path of Netflix regarding tv shows and movies. Both firms are increasing production of film, and now have their own productions.

If we were to categorize this type of competition, would it more closely resemble Hotelling's Model or Cournot?
It certainly isn't a Cournot scenario. That would require homogeneous products. In fact there's a ton of differentiation between the two products (Amazon prime and Netflix). They have different TV shows, different movies, different documentaries. 

I would argue that it is this differentiation that allows them to charge different prices. 

Hotelling's law is much more applicable. However I would argue that they do not maintain minimum differentiation. Yes they may share similarities in their web presence, and they may have adopted the strategy of producing content in house; but they also are so different that it is hard to say they are copying each other on almost everything.

Amazon prime offers reduced price shipping, music streaming, and reduced prices on some items on Amazon. Netflix has none of those things. 

That said, Netflix includes hard copies of DVDs as rentals included in their subscription, whereas you would have to buy the DVDs on Amazon. So even though some of their business is similar or matching, they mostly represent differentiated products.