The way that the Internet works hasn't changed, but what can be done with it has. The Internet works as the result of the transmission of packets through TCP/IP protocols, nothing more. The concept behind the technology that drives communication between two computers has not changed.
What has changed is the way it is being used. Web 2.0 is a change in that use. Much like how black and white TV broadcasts gave way to color, the way the Internet is used has evolved.
Previously, content on a webpage was highly static. The only one who could change it was the administrator or webmaster. Now, with Ajax and PHP, users are able to add their own content. This makes web pages more dynamic. This website is an excellent example of some of the concepts of Web 2.0, as individuals are able to add content as they see fit.
Web 2.0 is a change in the way the Internet is used, not how it is structured.
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