Superchats cause your message to be highlighted in chat. The more you pay for a Superchat, the longer your message will be highlighted. Donations are more like instant tips. They don't do anything special, but I get them instantly, unlike Superchats. Both Superchats and Donations will pop up on the stream for all to see. Private Donations are like regular donations except they do not pop up on the screen, and will not be acknowledged publicly. Do this if you don't want to draw attention to yourself and your donation.
To watch a replay of this stream as well as other past streams, check out this playlist: https://goo.gl/XGAxuS