What is the purpose of calling firebase.app() in react native firebase?

The two approaches are equivalent. The second one just relies on some hard-coded defaults, while the first is more explicit. This becomes especially apparent if you want to (for example) access to databases in a single app.

Our documentation explains this rather well, so I'll quote from there:

In most cases, you will only have to initialize a single, default app. You can access services off of that app in two equivalent ways:

// Initialize the default app
var defaultApp = firebase.initializeApp(defaultAppConfig);

console.log(defaultApp.name);  // "[DEFAULT]"

// You can retrieve services via the defaultApp variable...
var defaultStorage = defaultApp.storage();
var defaultDatabase = defaultApp.database();

// ... or you can use the equivalent shorthand notation
defaultStorage = firebase.storage();
defaultDatabase = firebase.database();

Some use cases require you to create multiple apps at the same time. For example, you might want to read data from the Realtime Database of one Firebase project and store files in another project. Or you might want to authenticate one app while have another app be unauthenticated. The Firebase SDK allows you create multiple apps at the same time, each with their own configuration information.

// Initialize the default app
firebase.initializeApp(defaultAppConfig);

// Initialize another app with a different config
var otherApp = firebase.initializeApp(otherAppConfig, "other");

console.log(firebase.app().name);  // "[DEFAULT]"
console.log(otherApp.name);        // "other"

// Use the shorthand notation to retrieve the default app's services
var defaultStorage = firebase.storage();
var defaultDatabase = firebase.database();

// Use the otherApp variable to retrieve the other app's services
var otherStorage = otherApp.storage();
var otherDatabase = otherApp.database();

Note: Each app instance has its own configuration options and authentication state.