The real difference between both is the role of the human. In artificial intelligence, machines acquire intelligence, intention and awareness. They, eventually, can replace human beings in certain tasks. Augmented intelligence, on the other hand, talks about machines complementing humans and making them better at accomplishing tasks.
This article has a very neat example:
If an AI system were pictured as an autonomously functioning robot, an HCI-inspired system might be more like the powered exoskeletons that will someday enable paraplegics to walk, or soldiers to have superhuman strength or endurance on the battlefield. AI aims to work alone; HCI aims to support and strengthen human actors.
You can also go through this post if you wish to delve deeper. Let us know your thoughts or any interesting articles on the subject in the comment section.