MEAN stands for Mongo, Express, Angular, & Node.
If you use Firebase, you would basically be replacing the MongoDB part of the MEAN stack with Firebase. You would still rely on Node / Express to be your server framework, and Angular to be your client framework. Firebase would then serve as your data source as opposed to MongoDB.
The initial configuration would be different. So if you started with some sort of generator ( such as Yeoman or a MEAN boilerplate ), you would probably need to go in and remove any references to MongoDB & Mongoose.
To answer the question, "Why use one over the other", it is really up to you. They are both good solutions.
Firebase seems to be better for real-time data, and in my experience is easier / faster to prototype with than Mongo. If you get into some heavy usage with Firebase, you will have to start paying for it.
MongoDB may have more documentation when it comes to using it with the MEAN stack.