Google acquired Quickoffice because it did a better job at handling Microsoft Office files than Google Docs. This was especially important for Google Apps users. Google started to work on porting Quickoffice to Native Client and integrating the application with Chrome OS. Now the mobile apps are free.
There's some overlap between Quickoffice and Google Drive apps, so offering two apps for editing documents, spreadsheets and presentations may seem strange. Quickoffice is a basic office suite that works offline, it lets you create and edit Microsoft Office files and annotate PDFs. You can store these files in Google Drive or on your mobile device. Files uploaded by Quickoffice can't be edited by Google Drive apps without converting them. Google Drive doesn't let you create or edit files when you are offline.
I find the Drive editing interface more intuitive and easier to use, but Quickoffice offers more features. When you open a file in Google Drive for iOS, the application first needs to upload the file to Google Drive. If you try the same thing in Quickoffice, the application opens the file and lets you edit it. I hope that the two apps will merge at some point.
Even if you don't need Quickoffice, it's still a good idea to install the app: you'll get 10GB of free Drive storage for 2 years. "If you sign in to your Google Account from the new Quickoffice app for Android or iOS by September 26, 2013, for two years, an extra 10GB of Google Drive storage will be added to your account in the next few weeks." If you don't get the extra storage, remember that will be added "in the next few weeks".
Here are the download links:
* Quickoffice for Android (phones and tablets, Android 2.2+)
* Quickoffice for iOS (iPhone/iPad - requires iOS 6+)
Those who paid for Quickoffice will probably be disappointed: no more updates for their apps, no iOS7 bug fixes. "Quickoffice Pro and Pro HD are not fully compatible with iOS 7 and some features may not work. After March 31, 2014, all cloud storage services will become unavailable on Quickoffice Pro and Quickoffice Pro HD," informs Google. Fortunately, "if you purchased Quickoffice Pro or Pro HD on or after June 17, 2013, you are eligible to receive 50GB of free Google Drive storage (with proof of purchase)."
(Quickoffice Pro and Quickoffice Pro HD used to cost $14.99 and $19.99. Pretty expensive, indeed.)