According to my understanding, module and controller is different.

1. Controller

Controller is used in view, handling the event triggered in view. The most explicate feature is to extend “sap/ui/core/mvc/Controller”.

Like this:

sap.ui.define([
   "sap/ui/core/mvc/Controller"
], function (Controller) {
   "use strict";
   return Controller.extend("sap.ui.demo.walkthrough.controller.App", {
      onShowHello : function () {
         // show a native JavaScript alert
         alert("Hello World");
      }
   });
});

Step 5: Controllers – Documentation – Demo Kit – SAPUI5 SDK (ondemand.com)

2. Module

Firstly, module is a normal principle for javascript. You can see this toturial: https://sapui5.hana.ondemand.com/#/topic/50579ddf2c934ce789e056cfffe9efa9

2.1 Usage

• Clean your code by module.
• Use module in your fragment. For fragment, you can assign a controller to the fragment when it’s loaded. Or you can use import module in fragment and use it like the below code:

<core:FragmentDefinition
	xmlns="sap.m"
	xmlns:core="sap.ui.core">
	<Dialog core:require="{ handler: 'sap/ui/demo/walkthrough/module/HelloDialog'}"
		id="helloDialog"
		title="Hello {/recipient/name}">
		<Button text="Button in Dialog"
				press="handler.onRaiseMessage"></Button>
	</Dialog>
</core:FragmentDefinition>

In some application generated by Fiori Element, it will be convenient to use the module in fragment. Since the controller and view are invisible.

2.2 Example

• Use sap.ui.define to define module.
• Import library.
• Define private method.
• Define public method(In the return statement).

sap.ui.define([
    "sap/m/MessageToast"
], function (MessageToast) {
    'use strict';

    var privateMethod_1 = function () {
        MessageToast.show("Active private method");
    };

    return {
        onRaiseMessage: function (oEvent) {
            privateMethod_1();
        }
    }
});

There are some points needs your attention:

• The public method cannot call another public method, it only can call private method. I am not clear the reason, if you know that, feel free to add comment.
• If you debug the statement “MessageToast.show(…)”, you will find that this refers to the view controller not the module. You can use it to operate model or do other things.

Example: https://github.tools.sap/I506216/UseModuleInSAPUI5.git