Gmail uses the pointer on mouse over of a button,
while many of other websites uses a hand
So I am wondering which kind of cursor is more friendly to users (on web applications), or it just doesn't matter?
Google has added this differing in hovering feedback to make a visual distinction between navigational elements and action elements in the UI. It's really to distinguish the semantics between actions like Compose a new mail and Open email.
So that is basically the strategy behind the behaviour. The reason however, how they feel that this will improve the user experience, I'm not really sure of.
What's important is for the user to get a feedback that will let him know that what he is pointing at can be clicked. Feedbacks can be various:
If you look closely at the google search button, it will change color when you hover it (even though the change is not that noticeable).
Normal Google search button:
Hovered Google search button:
I think the hand on hover is as you pointed out very very common metaphor for actions.
I'd say it's part of a users vocabulary, it's what everybody learns when using the web so it makes sense to use it if you want to optimise for usability.
Like @gotson said, visual feedback would also be appropriate.