Many people commonly use the words "Internet" and "Web" interchangeably, in particular, the two terms are both translated as 网络 in Chinese. However, they are not synonymous, in fact, the Web is an application running atop the Internet.
Linguistically, 'Internet' means between networks (net), as the definition of prefix 'inter-' is 'between or among', for example, 'inter-national' (between nations), 'inter-stellar' (among stellars), 'inter-relation' (mutual relation).
In contract with 'inter-', prefix 'intra-' means within or into something. For instance, 'intra-galactic' (in a single galaxy), 'intra-day' (within one day), and 'intra-net' (within a network).
But what is a network? A network is just some things or some people connected to each other, and the 'Internet' is just a network of computers connected in a wired or wireless fashion. Regardless of the transmission media (wire or radiowave), different computer networks follow the same rules. The set of rules used for connecting computers and networks is referred as Internet protocol suite, also known as TCP/IP stack.
In simple words, IP (Internet Protocol) gives each computer an identity (IP address), just like the unique ID number we have in a society. And TCP (Transmission Control Protocol) ensures messages exchanging between two IPs is done in a correct and secure way.
TCP/IP is the core of the Internet, and on top of TCP/IP, there are application level protocols (see the above figure). For example, FTP is a protocol for file transfer, SMTP is a protocol for electronic mail transport, and HTTP is the protocol that builds the Web.
The Web is a abbreviation of The World Wide Web, also abbreviated as WWW. While the Internet is organised to connect computers, the Web is mainly built to connect people, i.e., to exchange messages between people using the Internet.
The Internet is definitely not specifically designed for the web, however, the web is a increasingly dominant player in the Internet. More and more services are utilising the Web (the HTTP protocol) instead of using the Internet (the TCP protocol) directly. For example, video streaming services can use RTMP/RTSP protocol, which sits at the same layer as HTTP protocol, but nowadays, video streaming services are increasingly using the Web, which gives them a position above the HTTP protocol. Due to this change, web traffic constitutes almost all the Internet traffic, which explains why people tend to use the words "Internet" and "Web" interchangeably.
Lastly, to describe network related things, we have another prefix 'cyber-'. For example, cyber space, cyber crime, cyber attack, cyber extortion, cyber warfare, cyber terrorism, cyber violence, cyber bullying. But for some other things related to Internet, we prefer to use 'Internet' as an adjective, instead of using prefix 'cyber-', such as Internet Banking (or Online Banking), Internet Frauds, Internet Censorship, Internet Police, Internet Pornography.
If you are not sure about the choices between 'cyber-' and 'Internet', just consult Google. For example, Google returns 63 million results for 'cyber crime', and 28 million results for 'internet crime', which clearly indicates that 'cyber crime' is a more popular choice.