Thread.Start starts a new OS thread to execute the delegate. When the delegate returns, the thread is destroyed. This is quite a heavy-weight operation (starting and destroying a thread) so you typically only do it if the method is going to be long-running.
Delegate.BeginInvoke will call the delegate on a thread pool thread. Once the method returns, the thread is returned to the pool to be reused by another task. The advantage of this is that queueing a method to the thread pool is relatively light-weight because you don't have to spin up a whole new thread every time.
Control.BeginInvoke invokes the method on the thread for the control. UI components are inherently single-threaded and every interaction with a UI control must be done on the thread that created it. Control.BeginInvoke is a handy way to do that.