The Internet is a Network of Networks. Its the infrastructure that connects networks across the world, including the hardware computers, servers, cables and the software.
When you use an app on your phone, send an instant message, transfer a file directly from one computer to another via file transfer protocol (FTP) or simply send an email, you are using the Internet but not necessarily the Web.
World Wide Web is just another avenue for transmitting data over the Internet, in this case by entering a string of characters called a uniform resource locator (URL) into a browser. This URL describes the location of a document written in HyperText Markup Language (HTML), full of the links and silly cat GIFs you know and love.
If you typed www.doseofnews.com or www.cimages.me to get here, you just surfed the World Wide Web.