Personal Space: The best analogy I have for this is that it's like a personal planner. If you create a personal space for yourself, you have full control over it, and can do pretty much whatever you want to it. It's great to task management, personal notes, personal plans, documents, etc. Think of it as your own 'Home' in Confluence. This Space is for me.
Site Space: This is more comparable to a standard wiki, or in broader terms, a knowledge base. Generally these are used for information that lots of people will collaborate on and, potentially, modify. These Spaces are great for company-wide projects, clients, etc. This Space is for us.
This is just a very broad overview, but hopefully it'll give you an idea of the difference. Here's some more light reading, just in case you're curious!
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