Alright, time for a real comparison.
Instant communication. Both live chats and messengers are capable of this. The only difference is the way it's done. Live chat requires you to be online along with the customer representative for the conversation to happen. Messengers work the same way email does: send the message and wait for the response. No need to be online at the same time. A clear advantage for messengers.
Valuable for sales, support, and marketing. No doubt, both excel at what they do, with the difference being that messengers are at the beginning of being actively used for sales, while live chats have been at it for more than a decade. The same goes for support. As for marketing, messengers have a far better reach, being closely connected to social networks. Asynchronous communication combined with the ability to deliver content straight to the phone screen is invaluable for marketing campaigns. Lately, a lot of companies have started Live Chat Outsourcing as well. Live chats are a more traditional method of connecting a business for sales and support purposes, but messengers are clearly superior for marketing purposes.
Automation. Chatbotting like a boss can be done with just about any platform these days. With chat automation, messengers and live chats can ensure the business is available 24/7, reduce workload by taking care of repetitive tasks and more. Taking into consideration other strengths or weaknesses of both platforms, automation is equally beneficial for both messengers and live chats.
Integrations. Every service has its own pool of integrations with any given live chat/messenger platform. Services like Zapier and PieSync can help businesses integrate messengers.
Popularity. Messengers are the fad of the day, and almost everyone has one on their smartphone. Established messenger platforms aren't going anywhere and their functionality, API capabilities, and business opportunities continue to expand. Live chats, unfortunately, cannot step outside of their niche.