Packaged apps were Google's first attempt at creating apps for Chrome. Now they are about to be phased out in favor of Chrome Apps.
Packaged apps allowed you to run web apps as if they were installed, i.e. without the need to involve a server. They were superseded by a new improved spec for Chrome apps in the last quarter of 2013.
The new Chrome apps worked offline, ran in their own chromeless window and had a range of other enhancements. You could run a Chrome app anywhere you could run Chrome and, for the desktop, Google provided a special app launcher that made Chrome apps look even more like native apps.
I guess you could say that the writing was on the wall for the less powerful package apps and, from the end of June, no more packaged apps are being admitted into the Chrome Web store.
At the end of the year all existing packaged apps will be removed from the store's search and browse functions. At this point packaged apps can still be updated, but in June next year they will vanish completely. Hosted apps aren't affected.
The only option is to migrate to a Chrome app and Google has provided a guide to how to do it. This seems to be fairly easy. You have to update the manifest and set the Content Security Policy to something reasonable. Beyond that most of your existing code is reusable but you have to take account of a slightly more complex lifetime model and events.
What is really surprising is that after so very long it still seems to be difficult to implement a system for web apps that makes them the equal, or even nearly equal, to a native desktop app.