In simple terms, ResolveClientUrl returns a path relative to the current page, and ResolveUrl returns a path relative to the site root. Both methods are particularly useful when passing in a relative URL prefaced with the tilde (~) to indicate the application root.
Let's say we have an image in a layout directory, and a page which needs to pass a usable relative URL for the image to the client browser.
Here is what we would get if we called the two methods from our Default.aspx page.
There is more to it, of course, particularly since these are Control methods, not exclusively Page methods. It's pretty straightforward to see how paths relative to a page resolve, but not necessarily so obvious when the control is being instantiated in a UserControl or MasterPage.
ResolveUrl uses the Control.TemplateSourceDirectory property to do its job, and that property value is the virtual directory of the Page or UserControl that contains the current server control.
ResolveClientUrl returns a URL relative to the folder containing the source file in which the control is instantiated.