My personal experience is that sharing the whole screen is simpler and flexible (I can show anything without having to select another window), but it can cause privacy issues or make me look messy with icons around my desktop :)
On the other hand, picking a specific window is less flexible as I have to pick another window if the things I need to show up end in another window.
So, what do you typically do? Entire screen or window, why, what problems for each?
Thanks!