Neither of them can actually "win."
The Grim Reaper is traditionally just the personification of death, particularly as applied to humanity. Here's an amusing thesis set out by someone else. http://www.elfis.net/elfol5/e5death.html
It's amusing what one can find with a 5-second search.
In literature (for purposes of this thread, specifically modern science fiction and/or fantasy literature) death has also been personified as Death. Note that Death and the Grim Reaper are not alwaus used to refer to the same personification, but they can be.
Fritz Leiber, in his Fafhrd and the Grey Mouser stories, envisioned Death as a specific person. This Death knew that even he could die.
Similarly, in On a Pale Horse, Piers Anthony portrayed Death as an individual; in this case, a mortal chosen to take over the position of Death by killing the previous office-holder.
On a more abstract note, I observe that the concept of death
death (courtesy of Dictionary.com)
"Pronunciation Key (dth)
n.
The act of dying; termination of life.
The state of being dead.
The cause of dying: Drugs were the death of him.
A manner of dying: a heroine's death.
often Death A personification of the destroyer of life, usually represented as a skeleton holding a scythe.
Bloodshed; murder.
Execution.
Law. Civil death.
The termination or extinction of something: the death of imperialism. "
is inextricably linked to life. When all life is dead, regardless of the cause, then death too shall be gone.
If one insists upon personifying death in either form, then one can say that in the really long run they're both winners, but only in an abstract sense.