Setting module.exports allows the database_module function to be called like a function when required. Simply setting exports wouldn't allow the function to be exported because node exports the object module.exports references. The following code wouldn't allow the user to call the function.

module.js

The following won't work.

exports = nano = function database_module(cfg) {return;}

The following will work if module.exports is set.

module.exports = exports = nano = function database_module(cfg) {return;}

console

var func = require('./module.js');
// the following line will **work** with module.exports
func();

Basically node.js doesn't export the object that exports currently references, but exports the properties of what exports originally references. Although Node.js does export the object module.exports references, allowing you to call it like a function.

2nd least important reason

They set both module.exports and exports to ensure exports isn't referencing the prior exported object. By setting both you use exports as a shorthand and avoid potential bugs later on down the road.

Using exports.prop = true instead of module.exports.prop = true saves characters and avoids confusion.

Renee's answer is well explained. Addition to the answer with an example:

Node does a lot of things to your file and one of the important is WRAPPING your file. Inside nodejs source code "module.exports" is returned. Lets take a step back and understand the wrapper. Suppose you have

greet.js

var greet = function () {
   console.log('Hello World');
};

module.exports = greet;

the above code is wrapped as IIFE(Immediately Invoked Function Expression) inside nodejs source code as follows:

(function (exports, require, module, __filename, __dirname) { //add by node

      var greet = function () {
         console.log('Hello World');
      };

      module.exports = greet;

}).apply();                                                  //add by node

return module.exports;                                      //add by node

and the above function is invoked (.apply()) and returned module.exports. At this time module.exports and exports pointing to the same reference.

Now, imagine you re-write greet.js as

exports = function () {
   console.log('Hello World');
};
console.log(exports);
console.log(module.exports);

the output will be

[Function]
{}

the reason is : module.exports is an empty object. We did not set anything to module.exports rather we set exports = function()..... in new greet.js. So, module.exports is empty.

Technically exports and module.exports should point to same reference(thats correct!!). But we use "=" when assigning function().... to exports, which creates another object in the memory. So, module.exports and exports produce different results. When it comes to exports we can't override it.

Now, imagine you re-write (this is called Mutation) greet.js (referring to Renee answer) as

exports.a = function() {
    console.log("Hello");
}

console.log(exports);
console.log(module.exports);

the output will be

{ a: [Function] }
{ a: [Function] }

As you can see module.exports and exports are pointing to same reference which is a function. If you set a property on exports then it will be set on module.exports because in JS, objects are pass by reference.

Conclusion is always use module.exports to avoid confusion. Hope this helps. Happy coding :)