Thanks to @_Everaldo, I learned that the new Twitter for Mac does, in fact, have a "secret" alternate icon in the Resources folder.

If you would like to switch to this secondary icon you will need to go to quit Twitter.app > Remove Twitter from the Dock > Go to your Applications Directory > Right-click (Option + Select) the Twitter app and select Show Package Contents.

Navigate to Contents > Resources. In this directory, you will find files called "Icon.icns" and "Icon_2.icns".

Drag both these files out to your desktop and delete the originals. Rename "Icon.icns" to "Icon_x.icns" or whatever you want, then rename "Icon_2.icns" to "Icon.icns". Drag the newly renamed alternate icon back into the Resources folder and enter your computer's Password when prompted.

Go to the Force Quit menu (found in the  drop-down in the Menubar) and Relaunch Finder. Relaunch Twitter.app. Your new icon should be the alternate icon. 

I would recommend keeping the original in case you ever want to switch back.

Update: Found another alternate icon for Twitter for Mac on Dribbble.

Also, here's another really sweet little hidden feature of Twitter for Mac!