Community Q&A is designed to surface the questions and answers that have been asked on a page publicly. The Q&A thread enhances the host page. The Pubblebot will query all Public Q&A's to try to deliver an instant answer. Community Q&A is perfect for building micro-communities for every page of your site. Messenger on the other hand is designed to support direct messages from your users. The conversations are 1:1 and are never public. The Pubblebot only prompts from defined FAQs.
Messenger offers website messaging that you would actually want to use, every time.