What is .NET framework backward compatibility?

Question

We develop a windows application in .NET 4 with a database of MySQL. When we are about to deploy that application, we install the .NET 4 framework in our client, then when installing the .NET Connector for MySQL, it requires a .NET framework 3.5. But the .NET 4 is already installed. Wasn't .NET framework backward compatible?

We end up installing two .NET framework (3.5 & 4) which isn't small in disk size. What exactly does backward compatibility means?