What is the exact difference between Windows Azure and Windows IIS?

What you are looking for is actually a place to run your web application, Teo.

As you've found, you can do that in IIS if you have a server that is connected to the Internet. A way to get such a server is to either got to a hosting company or just use the Windows Azure cloud as you've found as well.

One of the simplest ways for you to do this right now and for free is to sign up for a Windows Azure trial account. As part of that account you get a basic, shared Windows Azure Website for free.

Here are the links you need:

(1) http://www.windowsazure.com/en-us/develop/net/tutorials/get-started/

(2) http://www.windowsazure.com/en-us/pricing/free-trial/

I would strongly recommend that you go through the entire tutorial (1) step-by-step before trying to do this with your own application. Before you start, sign up for a trial account (2). You will not be charged in the first month and you will not be charged if you stick with the free website.

Enjoy.

Comparing IIS to Azure is irrelevant. Those are two different concepts, which are vaguely related to each other. You lack some very basic understanding of what each one means, and I recommend you to go and read about each them from scratch.

IIS is indeed a web server application. That means, for example, that it can rout HTTP request and responds to and from the web site application that you have created.
To keep it simple, let's say that IIS can run on any Windows machine, which makes the machine a Web Server.

If you want to have your web site up and running, you need either have your own machine that acts as a web server, or either upload your web site application to some other machine.

Azure is a group of cloud services. One of the services is a Web Site Host, that allows you to use cloud computers to run the IIS that hosts your web site.
As part of the service, Azure will take care of installing and using the IIS server for you.

Bottom line, if you are going the Windows path, you will probably end up using Both Azure and IIS (unless you will want to self host your web site...)