The “web-based” solution, which runs on a remote server, is generally able to be reached by the general public through a web page. It constitutes a web page with a chatbot embedded in it, and a text form is the sole interface between the user (you) and the chatbot. Any “upgrades” or improvements to the interface are solely the option and responsibility of the botmaster.
Messenger Bots are created using the new Messenger API that allows a bot to send and receive messages. The Messenger Bots are essentially chat bots that you can talk to from the Messenger app. The conversations will of course be different than those you have with your Facebook friends. These bots are meant to help you get information for example you can ask the CNN bot to tell you give you the current headline news and it will fetch them for you.