@Phobos IP is an addressing scheme that the various devices that make up the internet (computers, routers, servers, etc) can use to send packets from one place to another. Applications are programs running on computers, some of which use the IP addressing system to allow the computers they're running on to communicate with other computers elsewhere in the world.
@Kagen/Vorash, the internet refers to physical infrastructure (literally wires carrying bits) and low-level protocols (like the IP addressing scheme) that enable computers to send packets to one another. The web uses this infrastructure to enable browsers to send and receive HTTP, which encodes webpages like the one you're on – it's an application running on top of the web via servers and browsers.
Also, the "wire" is often pretty different from the phone cables of yesteryear – a lot of it is fiber optic cable, which can transport literally millions or billions of times as much data (in the form of packets/second) as your average telephone line.