There is no difference in effect; in both cases the TargetType property will be set to typeof(Border)
The first version {x:Type Border} was needed in the first version of WPF because the compiler did not use the TypeConverter class to convert the string into a Type object and you needed to specify the TypeExtension class to do that for you.
The second version was introduced, if I remember correctly, with Silverlight and quickly found its way to the WPF compiler.
EDIT
My assumption on the TypeConverter class was wrong; this is implemented by the FrameworkElementFactory:
From the documentation:
Type Properties That Support Typename-as-String
WPF supports techniques that enable specifying the value of some
properties of type Type without requiring an x:Type markup extension
usage. Instead, you can specify the value as a string that names the
type. Examples of this are ControlTemplate.TargetType and
Style.TargetType. Support for this behavior is not provided through
either type converters or markup extensions. Instead, this is a
deferral behavior implemented through FrameworkElementFactory.Silverlight supports a similar convention. In fact, Silverlight does
not currently support {x:Type} in its XAML language support, and does
not accept {x:Type} usages outside of a few circumstances that are
intended to support WPF-Silverlight XAML migration. Therefore, the
typename-as-string behavior is built-in to all Silverlight native
property evaluation where a Type is the value.