in the early days of SharePoint Online (and SharePoint on Premise without the Office Apps Server installed)  you could not co-authoring with all Office documents online. For example, if you have had a word document, then you have had to open and edit it with the native Word 2013, if you want to co-authoring with other users.

Now, you only have to watch Excel

Here are the possibilities with Word and PowerPoint

Edit a document User 1	Edit a document User 2	Co-Authoring
Word Online	Word  Online	Yes, best user experience
Word Online	Word started from Browser	Yes
Word started from Bowser	Word started from Browser	Yes
Document opened from OneDrive for Business  (local file)	Word Online	Yes
Document opened from OneDrive for Business  (local file)	Word started from Browser	Yes
Document opened from OneDrive for Business  (local file)	Document opened from OneDrive for Business  (local file)	Yes

Excel

Edit a document User 1	Edit a document User 2	Co-Authoring
Excel Online	Excel Online	Yes
any other combination	Excel Online	NO

This time you may only Co-Authoring with Excel Online !!

	on the right side of your browser, you see your co-authors in Word online
	with Word Online you see the others Co-authors writing

 

With SharePoint Online you may co-authoring and you may decide, which program you will use. You will have the best user experience, if you collaborate with the Online versions. But in some situations you will take the native Office programs, f.e. if a specific function is not available in the Online Version.

Keep in mind with the native Office programs and “big” documents, only the changes (deltas)of the document will synchronized. And this is a big advantage of OneDrive for business.

Conclusion:

Microsoft should close the gap on Excel, then the user may decide, which is the best way to Co-Author with other users.

My wish-list Microsoft my implement in the future versions

• save parts of my document, nobody may change or delete
  (Only the last user, who ends the document writes a new version, if version history is on)
• implement a log, which user do what
  (in word you may do that with review an track changes – but only with the native Word)

Co-Authoring Link