Unable to install Xcode using dmg file downloaded from Apple.com

As Adam stated, Xcode 4.3 switched over to being a self-contained application. I would suggest uninstalling the old version of Xcode as he stated, then just copying the new .app over into your Applications folder. Finally, delete the whole /Developer directory.

However, Apple decided to make the mistake of not updating the paths for the development tools. So, you need to update those paths manually. If you've put Xcode in your Applications folder, you can run the following in Terminal to switch the paths over to your new app location:

sudo xcode-select -switch /Applications/Xcode.app/Contents/Developer

If you need to check the current path, you can do the following:

xcode-select -print-path

It may help to manually remove the Launchpad databases from your home directory. The databases should be located at ~/Library/Application Support/Dock...The files may have UUID names (long string of hex chars)...Removing those files, performing a logout/login (or a reboot) should allow those files to be rebuilt from the exiting applications in /Applications.

You can get into the Library folder by holding down the "Option" key on the keyboard, clicking on the "Go" menu (when you're in the Finder), and choosing "Library".