<< . . They probably are the same image if the URL is the same and there is no code to detect whether you're on a mobile device or not . . Detection as to whether the site is being accessed from a traditional computer or a mobile device isn't always easy. >>
I don't understand. An app is made for smartphones — the user downloads it to the smartphone, not to somewhere else — installs it on the phone — runs it on the phone. There's no need to detect if it's a mobile device: it's a mobile app, it can only be on a mobile device.
Websites and their sister smartphone apps are usually completely different, and I'm assuming the reason is that an app's appearance must conform to phones' aspect ratio, limited real estate, and desire for quick loading. I checked out numerous sites and only a few, such as Google maps, have identical appearance on web site and app. As I mentioned before, I also don't want the app to have a background image, it serves no purpose there (whereas on the website it looks gorgeous (in my not very modest opinion ) and greatly enhances the appeal of the site.
<<You want to try and have as few versions of each image as possible: the more you have, the more you have to change when you don't like something or want to change the look and feel.>>
My first response was "Thumbs up to that! Especially since I've come to like my image for the app better than my old HTML image", but after some tests I find that shaping the gamecontainer with the aspect ratio of smartphone screens (16:10 Android, 16:9 iPhone) just doesn't look good on the web page. It's important that the app's gamecontainer neatly fill the entire phone screen viewport, and equally important that the web page have a gamecontainer aspect ratio and size/locationwhich looks good on typical computer screens, so I'm stuck with having to have a separate gamecontainer for each. But I don't mind, I'm getting pretty adept with Photoshop so making changes no longer takes me a lot of time.