The problems, as I see them:
1. The two parts (DVD & streaming) had different characteristics (bigger collection vs. instant gratification) that balanced each other. This synergy is now gone.
2. The attack surface for other companies has just doubled. Netflix's DVD handling is super streamlined (they have special sorters in postal centers) but there are other strong players like Redbox here. The streaming biz is nothing unique, Amazon can easily get to that level of collection in a short time. Divide and conquer? They just did it to themselves!
3. As others have pointed out, the reviews, and user histories will be split. From a subscriber perspective this is most undesirable, now I have to deal with two sites.
4. The quality of recommendations will suffer with less data (they weren't great to start with) after the split.
5. How will this affect developers using the Netflix API?
6. What will happen to Netflix's original content creation efforts, this will be a huge distraction to those. And they are vital.
Overall, this has a heavy handedness to it that makes you wish for the comfort of the likes of Amazon.
EDIT: This also makes my suspicions that Amazon is gearing up to a massive upgrade of their offering soon. Netflix probably tried to focus on streaming and stay ahead in the game. How? I don't know.