Difference between a class and a module

The first answer is good and gives some structural answers, but another approach is to think about what you're doing. Modules are about providing methods that you can use across multiple classes - think about them as "libraries" (as you would see in a Rails app). Classes are about objects; modules are about functions.

For example, authentication and authorization systems are good examples of modules. Authentication systems work across multiple app-level classes (users are authenticated, sessions manage authentication, lots of other classes will act differently based on the auth state), so authentication systems act as shared APIs.

You might also use a module when you have shared methods across multiple apps (again, the library model is good here).

╔═══════════════╦═══════════════════════════╦═════════════════════════════════╗
║               ║ class                     ║ module                          ║
╠═══════════════╬═══════════════════════════╬═════════════════════════════════╣
║ instantiation ║ can be instantiated       ║ can *not* be instantiated       ║
╟───────────────╫───────────────────────────╫─────────────────────────────────╢
║ usage         ║ object creation           ║ mixin facility. provide         ║
║               ║                           ║   a namespace.                  ║
╟───────────────╫───────────────────────────╫─────────────────────────────────╢
║ superclass    ║ module                    ║ object                          ║
╟───────────────╫───────────────────────────╫─────────────────────────────────╢
║ methods       ║ class methods and         ║ module methods and              ║
║               ║   instance methods        ║   instance methods              ║
╟───────────────╫───────────────────────────╫─────────────────────────────────╢
║ inheritance   ║ inherits behaviour and can║ No inheritance                  ║
║               ║   be base for inheritance ║                                 ║
╟───────────────╫───────────────────────────╫─────────────────────────────────╢
║ inclusion     ║ cannot be included        ║ can be included in classes and  ║
║               ║                           ║   modules by using the include  ║
║               ║                           ║   command (includes all         ║
║               ║                           ║   instance methods as instance  ║
║               ║                           ║   methods in a class/module)    ║
╟───────────────╫───────────────────────────╫─────────────────────────────────╢
║ extension     ║ can not extend with       ║ module can extend instance by   ║
║               ║   extend command          ║   using extend command (extends ║
║               ║   (only with inheritance) ║   given instance with singleton ║
║               ║                           ║   methods from module)          ║
╚═══════════════╩═══════════════════════════╩═════════════════════════════════╝

I'm surprised anyone hasn't said this yet.

Since the asker came from a Java background (and so did I), here's an analogy that helps.

Classes are simply like Java classes.

Modules are like Java static classes. Think about Math class in Java. You don't instantiate it, and you reuse the methods in the static class (eg. Math.random()).