According to the apps' profile page on the Google Play website, Google Docs has been designed to "create, edit & collaborate with others on documents from your Android phone or tablet".
Quickoffice on the other hand enables users to "create and edit Microsoft Office documents, spreadsheets and presentations", while Google Drive to create and edit documents that appear to collaborators in seconds.
From what I can tell, Google Docs focuses on the collaboration aspect. While you can create and edit documents in a similar fashion as you can do with Google's two other applications, they don't support the collaboration features that Google Docs does.
You get real-time collaboration with Google Docs which means that you see what others do when they work on a shared document, spreadsheet or presentation.
This includes adding comments to documents, and responding to comments that other users have left. To answer to comments, you need to tap on the settings icon at the top right when the document is open to select the comments option from the listing.
Google Docs displays all comments to you. When you tap on a comment you get options to reply to it or edit it if it was your own.
Google Docs uses Google Drive as its online storage solution, but unlike the latter can make available documents offline on the device.
You can make documents available offline with a tap on the (i) icon next to them and the selection of "keep on this device". This saves an offline copy of the file on the device so that it can be opened and edited without Internet connection.
The same menu displays additional information including users with access, options to share the document with additional users, and the last modification and opening information.
Documents that you make available offline highlight the fact with a small pin icon underneath the document title in the document listing.
The app lists documents by date only which can be problematic if you prefer to organize documents in folders instead and have troubles finding documents. The search can help with that, but only if you can remember the document name.
You can display the Google Drive listing however which displays the folders here. That too may be confusing however as you cannot really tell if you are still in the Google Docs app or the Google Drive app, or somewhere in between.
If you collaborate on Google Docs a lot either in real-time or using the comment system, then you will certainly appreciate the Google Docs app. While the functionality could have been integrated into Google Drive as well, some may like the fact that it does one thing well instead. It appears that the opening is snappier as well, but I do not have hard data to back this up.
Still, Google should make it clearer how those apps differ, as it is somewhat difficult to tell right now.