Why does one of two lights stay on when they share a neutral?
Without knowing exactly what you've got going on, it's impossible to say for sure what you have to do. It's totally possible that there are no "neutrals" in the box at all, depending on when the house was wired (and by whom). 

Here's what the schematic should look like...



Basically, any wire that comes "after" a load, and creates a low resistance path back to the "source" is a "neutral". Which means you can further label the diagram as such...



The dimmer requires a "neutral", because it itself is a load. So it requires a low resistance path back to the "source" as well.

A wire coated in white insulation, does not make a wire a "neutral".