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I often hear people say "the Net" when they really meant "the Web." This analogy is helpful in explaining the difference between the Internet and the Web, or between the various Internet protocols.
The Internet runs on a protocol called IP, literally this means "Internet Protocol". IP is a very simple protocol which means it's a very fast protocol for devices to process. TCP/IP is one of the protocols that exist ontop of IP and uses IP to create a reliable connection between two addresses.
On top of TCP/IP are the other various services and protocols. But first and foremost, to have the Internet, you have to have IP and TCP/IP. Think of TCP/IP like a road, and the various services that run on top of TCP/IP like the vehicles that travel on that road. IP is the stone and earth that support the road, the bridges and tunnels that allow it to go to impossible places.
Before you can travel at all in a vehicle, you have to have a road to drive on. That's TCP/IP. In the same way that a road doesn't care what vehicles are riding on top of it, TCP/IP doesn't care what services are run on top of it. It merely provides a place for transportation to occur. The road, which is TCP/IP, is therefore what we should think of when we say "the Internet."
Lots of different vehicles can ride on a road: cars, trucks, vans, buses, motorcycles, and even horse-drawn carriages. People drive different kinds of vehicles for different purposes: speed, comfort, the number of people that need to be moved, and so on. Those different vehicles are like the different services that people use on the Internet. Instead of cars, trucks, and vans, we would instead refer to the Web, email, and instant messaging (IM).
Most of the time, vehicles don't have to worry about the road they're traveling upon. It's just there, and it just works: no huge potholes, no restrictions, no places where the road is gone or leads to nowhere. Likewise, most of the time people just use the Web or email or IM without having to worry about TCP/IP. That's a good thing, and that's part of the reason that the Internet works as well as it does.
Every once in a while, something happens to the roads we drive on. A part of the road is blocked, or washed away by extreme weather, or damaged, and traffic on that road suffers. Vehicles can't use the road any longer, or it takes a lot more time to traverse it.
Every once in a while, something happens to the Internet, causing TCP/IP not to work. A cable in the ground is snapped by construction workers, or a piece of equipment that helps route Internet traffic breaks, and suddenly the services that depend on TCP/IP don't work as well as they should. The Web becomes slow, or your email isn't delivered, or it appears that your friend isn't available on IM even though he really is.