Okay, I figured out: my machine's port 80 was being used by IIS, not by Skype. Here is how I found that:

Left-click the WAMPManager icon to reveal the menu. Click Apache ► Service ► Test port 80. This will launch a command window and display some information about what, if anything is using port 80. You can find more details at http://forum.wampserver.com/read.php?2,122527.

Configure Apache to listen on different port
Since I wanted to keep running IIS on the port 80, I decided to run my WAMP Server on a different port. So here is how I did it.

I had to change http.conf file which in my case is in folder C:/WAMP/bin/apache/apache2.4.9/conf/. Open it (with Notepad for example) and search for 80. I had to change the following three lines on different places:

Listen 0.0.0.0:80         # Changed to Listen 0.0.0.0:81
Listen [::0]:80           # Changed to Listen [::0]:81
ServerName localhost:80   # Changed toServerName localhost:81

Now I have got the green icon of WAMP Server on Windows 8.

More details on http://forum.wampserver.com/read.php?2,13744.