Q:

What can you do in background scripts that you can't do in other js files?

*By other js files, I mean the files you include in popup.html.
The following code works, so why should I use a background script?
Content script
chrome.extension.onMessage.addListener(
  function(request, sender, sendResponse) {
  //Some code
  }
);

Script included in popup.html
chrome.tabs.query({active:true,windowId: chrome.windows.WINDOW_ID_CURRENT},
  function(tab) {
    chrome.tabs.sendMessage(tab[0].id, {method: "someMethod"},
    function(response){
      //Some code
    });
  });

This

A:

Background pages live as long as the Chrome browser is active (and even longer if the "background" permission is set). Popup pages are only active when the badge's popup is open. The popup can only be opened by the user.
The background page's document is never visible, whilst the popup page becomes visible on click of the badge's button.

Besides that, there's no difference between background pages and popup pages. They run in the same extension's process, and have access to the same set of APIs.
If your extension only needs to be active while the popup is active, you don't need a background page. To save the state of the popup, just use the synchronous localStorage or the asynchronous chrome.storage API. When the variables you use are too complex to be stored using either API, a background page may be useful.
One example where the use of a background page is beneficial:
Imagine an extension that downloads a huge text file from a server. The creation of the resource is very resource-intensive for the server. Technically, everything can be done from within the popup. However, offloading the task to the background page allows the user to do other tasks while the file is downloading (if you use a popup only, the download will stop when the user closes the popup).

Though you didn't ask, I'd like to make you aware of event pages. They are similar to background pages, with one difference: Event pages are automatically closed when the extension is idle. In other words, event pages are only active when needed! By doing this, your extension will profit from the advantages of background pages without needlessly wasting the user's memory.
My last example is also a perfect example of when an event page has to be used. Besides doing the http request on behalf of the popup, the background page does nothing. If you use an event page instead of a background page, you get the best of both worlds: the popup page can be closed without interrupting the download, and the extension will not waste memory.
Documentation

Learn more about Background pages and Event pages
"Popup" in this answer refers to the optional panel of the chrome.browserAction or chrome.pageAction API, set by declaring the "default_popup" key in the manifest file, or programatically using the setPopup method.

