They both use the addView() method along with various windowManager methods. The two are similar in that regard.
Dialogs seem to come with more built-in features for interaction, such as handlers and buttons already included in the base class, while PopupWindows come with more built-in methods for positioning them about the screen.
I think that each of them can do exactly the same as the other, but choosing between the two will be a matter of convenience to the programmer with regards to how you want to use the Object. I'm not a phD in computer science, but I do not think there is a significant difference in processing time between the two based on what I saw in their respective class definitions.
My advice: If you want to have greater control over where your View appears on the display, use a PopupWindow. If you want to add more control and feedback between your View then use a Dialog. If you, like me, want master control over everything, I would suggest a PopupWindow since it has fewer user-evident default methods to override.
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