Websites have pages. Does software? I'm mainly thinking of smartphone apps.
'Interface's is probably technically correct, but 'interface' could equally refer to sub-menu that appear within a given 'page'.
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I would go with the same i.e. page. This would be making it easier to know what you refer to since it's a convention on web sites. User Interface refers more to the entire application, which contains several pages - like a web site also have several pages. Furthermore you don't have to distinguish between web apps or native apps.
Scribd.com has published the article Terms Common In Mobile Development where you can read under the sub heading metaphors that:
Common metaphors include, tapping playback controls, dragging, flicking, or swiping objects, slidingon/off switches, flicking through pages or photos, and spinning picker wheels to make choices
I'd say it's safe to use a page as the name of a view or screen in any web or native mobile application.
Typically software developer parlance is "screens", "pages" is more about websites. Windows is also used.
In applications where there are sub-sections there are often separate screens that might be more accurately referred to as "menus" and "modules", depending on if they open a new screen or window. This is true of mobile app development, and even web apps - but not web sites, where pages is used.
For immersive enviroments there are rooms, areas and levels and other more specifically descriptive terms for new pages or screens.
So there are templates, and there are pages. I think this can be decent way for the naming in this field http://bradfrost.com/blog/post/atomic-web-design/