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The difference is that with normal shared hosting you are buying web space on a server. You can't install your own software and you have lower limits for your website. This is fine if you're just running a website that doesn't have huge amounts of traffic and don't need custom software. It's a lot cheaper to use shared hosting for most websites and a lot easier to use since you'll usually have access to a control panel such as cPanel.
If you have a VPS then you can install anything you wish however unless you have a managed VPS you're left to install everything yourself. This means purchasing your own licenses for cPanel, DirectAdmin etc and then installing everything and keeping it secure. The benefit is that although it's still a shared service you'll have your own slice of the servers CPU, RAM etc and have a lot more control over everything.