Some example - I am using Poncho bot that sends me weather update every day in morning and evening. So that way I don't need to install other app or need to open site and check everytime.
Same way, Machaoo bot sends me match score update on a regular interval.
1- Being where your customers are. If you are planning on leveraging a chatbot as opposed to a mobile app or website, having some kind of presence of Facebook messenger could increase user engagement because of the lower amount of friction from switching or downloading new apps to interact with a business. It's especially relevant when you consider that social platforms are used for customer support more and more so augmenting this channel with some kind of automated support option makes sense from an ROI perspective.
2. Simpler user experiences. Now this one is definitely idealistic, and where a lot of hype is coming from. But, the general idea is to take websites and mobile apps and create a single interface into the content/services where the hope is that users will have a lower learning curve because they can use everyday language. In reality, bots that I've seen are mostly text based IVRs that confuse the users, and don't make experiences any more enjoyable.
3. Personalization and NL analytics. Assuming your chatbot allows for free form text. You could potentially build more comprehensive profiles of users by mining their conversations and behavior. You could certainly do this today with web and app metrics, twitter posts, support emails, and transcribed voice calls, but now you have a Facebook or social profile tied to a customer which is another dimension of data you can use where in the past you may have had no linkage between your app accounts vs a users social profile.
By no means a comprehensive list, and if you incorporate voice recognition on top of a "chatbot", you can provide an easier way for users to interact with systems when they may not be able to use their hands (driving, lost an appliance remote, running)
For the users the advantage is that they don't have to install anything. The disadvantage is that it's a more crude UI than a full fledged app or a modern mobile website but that can be hidden by a good design.
However a conversational UI mostly flows in time with little memory (scrolling and searching are possible but not user friendly), whilst apps/desktop/websites mostly spread in a 2D space. The design of the interaction is very different. Just try some mockup tools for chatbots and you'll find it out quickly.
There are advantages for developers too: it's easier to develop a chatbot than an app (unless you go very deep in AI), as much as it was to develop a web app than a desktop one. It costs less, it delivers earlier, it can be updated 100 times per day without app store approvals.
Edit: s/full fledged API/full fledged app/