The most frequent feedback from community members of the website Reblog button is that it immediately publishes the reblog to your website when you clicked the button. That didn’t allow you, the reblogger, to preview your post, add your own site’s categories and tags, save your reblog as a draft post, schedule your reblog, add your own featured image or customized social media message and whatever other options are available in the Editor.
However, you can do all those things with the Reader‘s Share button and the WordPress.com PressThis bookmarklet since both of those open the Post Editor.
(Link to the original forum thread that inspired this post.)
One valid point that a community member raised in that thread is the website Reblog button doesn’t allow the reblogger to change any of the originally blogged text. I have my own thoughts about anyone who would do that and what doing that does to that person’s authority on a subject.
Staff have updated that forum thread to indicate that the missing Reblog button is a result of removing some deprecated code, which also affected the website Reblog button. The Devs are currently working on a solution and testing potential fixes.
WPcomMaven 
Thank you very much for the tip about reblogging from the Reader! This is incredibly helpful. I don’t use reblog all that much but when I did, I used to publish then go back into the post to change featured image, add tags, etc. This feature allows me far more control. Thanks again! 🙂
You’re very welcome. Staff advised a little while ago that the website Reblog button has been fixed so now you have another choice. 😊 Thanks for stopping by.