The reason you are seeing this behavior is due to the different runtimes available for apps to use in Windows Phone 8.1.
Since Windows Phone 8.1, there was a major change to the lifecycle of apps. Tapping back from the main page of an 8.1 WinRT (Universal) will not close an app. This is also the same lifecycle you see in Windows 8.1 apps.
However, if the app uses WP 8.0 (Silverlight), the app will close by tapping back from the first page of an app (unless the dev screwed up and has a circular navigation happening, in which it will go to another page in the app instead of leaving the app).
There is a variety of choices for devs today; 8.0 Silverlight, 8.1 Silverlight and WinRT (Windows 8.1 model).
Just to be clear, apps never run in the background on Windows Phone. They go into a suspended state. If the app has been suspended for a long time, been pushed down too far by other apps or if you close it manually they go into terminated state.
When you go back to a suspended app, you'll see the "resuming..." notice. This is when the developer loads the data that was saved when it was suspended. So from your perspective, it was never closed.
-A small summary of what does run in the background-
What does run the background are "background tasks" which are small snippets of code that run independently of your app, even when the app has been terminated. That's why you have the option in settings to disable background tasks.