The only difference is that if you are using user Controls into a control from Same Assembly then Name will not identify your control and you will get an error " Use x:Name for controls in the same Assembly". So x:Name is the WPF versioning of naming controls in WPF. Name is just used as a Winform Legacy. They wanted to differentiate the naming of controls in WPF and winforms as they use attributes in Xaml to identify controls from other assemblies they used x: for Names of control.
Just keep in mind dont put a name for a control just for the sake of keeping it as it resides in memory as a blank and it will give you a warning that Name has been applied for a control buts its never used.