Internet Explorer has two versions on Windows 8/RT:
(a) The desktop version of IE supports plugins and accelerators. This is the version that you launch from the Windows 8/RT desktop.
(b) The Windows Store app version (with "Modern" or "Metro" UI) of IE is touch-optimized, full screen, and plugin free. This is the version that you launch by tapping/clicking the tile in the Start screen.
See the Microsoft documentation titled Get ready for plug-in free browsing.
with Windows 8, IE is one web platform that provides two browsing
experiences: the new IE in the new Windows UI optimized for touch
devices, and the traditional browsing experience of Internet Explorer
for the desktop. As a Windows app, Internet Explorer runs
without plug-ins so that you have a clean, fast, and secure web browsing
experience, while also providing a native Adobe Flash player. "
Similarly, Adobe Reader also has two versions on Windows 8/RT.
(a) The desktop version is called Adobe Reader XI. It is a mature product that has been available in the market for the past 20 years and offers a rich set of features. It supports in-browser viewing of PDFs by installing the browser plugin to the desktop version of IE. However, the downsides of the desktop version are the incompatibility with the Windows RT operating system and security vulnerabilities. In general, the desktop version is less secure than the Windows Store app version.
The Windows Store app version is called Adobe Reader Touch. It is a
relatively new product that was introduced a year ago. It does not
support in-browser viewing of PDFs because of the security restrictions
that are imposed on all Windows Store apps. It is compatible with both
Windows 8 and RT operating systems.
Because the Windows Store app version of IE is designed to be plugin-free, no other desktop/Windows Store apps (hence, not even Adobe Reader XI) can install additional plugins (other than the Flash support that is built-in by default).
If you want the in-browser PDF viewing support, you need to use the desktop version of IE and the desktop version of a PDF viewer like Adobe Reader XI.
This is the application architecture decision that Microsoft has made deliberately. It's not a bug or oversight.
Microsoft has the Windows Store app version of PDF viewer called "Reader". It may be possible for Microsoft to add the in-browser PDF viewing support using Reader (just like Google Chrome does) in a future release of the Windows Store app version of IE. But it did not happen in the newer version of the operating systems: Windows 8.1 and Windows RT 8.1.
The information source is the Adobe Forum: http://forums.adobe.com/thread/1327187