The major difference between these two kinds of programming is where the program runs. Client-side programming runs on the client's computers while server-side programming runs on the company's own servers.
Client-side programming (which includes Java, for example) is more flexible than server-side programming. It can respond to and change with things that users do with their computers or to other things like the time of day where the client is. However, it is less secure than server-side programming and it is harder to deal with any bugs that may arise.
If you follow the second link, you can listen to a brief explanation of this difference.