Some people are annoyed by the Facebook Messenger app because the official Facebook app no longer supports messaging and forces people to use the App. It is Facebook’s way of getting more people to use the Messenger app so they can compete with other social messaging apps.
The best alternative to Facebook Messenger is to use the mobile version of facebook.com from your smartphone which still supports chatting with people. Just open your browser and go to Facebook.com and you can chat once you click the messages tab.
I don’t think there are any 3rd party messaging apps that can allow you to chat with your Facebook friends. If I do come across any such apps, I will be sure to list them here.
What annoys me the most about Facebook Messenger app is all the silly sounds it makes when someone is writing a message or when I am writing and sending messages. It is overkill. Another annoyance is the fact that chat heads stay up and get in your way while you are browsing or trying to use your phone. I am not the first to complain about this either.
Also do consider reading the Phone Arena article on how to circumvent Facebook Messenger.