Windows Metafile (WMF) is a graphics file format on Microsoft Windows systems, originally designed in the early 1990s. Windows Metafiles are intended to be portable between applications and may contain both vector graphics and bitmap components.
Essentially, a WMF file stores a list of function calls that have to be issued to the Windows Graphics Device Interface (GDI) layer in order to display an image on screen
In 2007 Enhanced Metafile (EMF) a newer 32-bit version with additional commands appeared. EMF is also used as a graphics language for printer drivers. The last(?) version of EMF, 4.0, appeared in 2008.
With the release of Windows XP, the Enhanced Metafile Format Plus Extensions (EMF+) format was introduced. EMF+ provides a way to serialize calls to the GDI+ API in the same way that WMF/EMF stores calls to GDI.
So essentially EMF is a newer better updated version of WMF. I suppose WMF is still around for back portability.