If you hard-code the type, like PartialFunction[Any,Nothing], you cannot restrict your function to take a more specific parameter than Any.
By using a generic type parameter, you can end up with a more flexible satisfying all cases and especially making the function safe.
Let's assume you want a function aiming to take an Animal as parameter and returning an Integer.
Let's assume that function is declared as being:
def myFunction: PartialFunction[Any,Nothing]
Firstly, PartialFunction would not be specialized to Animal at parameter side but to Any. What about if I pass a Human as parameter...., it would pass.. What about safety?
Secondly, If this function is declared as returning Nothing, you can't return from it any value but Nothing! Indeed, Nothing subclasses all classes in Scala.
This leads to the known rule that return type parameter must always be covariant in order to make a function interesting, not the case with Nothing.
In fact, Nothing is interesting only when dealing with the empty method of PartialFunction. Logic since an empty PartialFunction by definition involves nothing to return and should be forced to do it :)
You would ask: "So why don't we change the return type to Any?"
Answer: Because you'd lose all the benefit of generic erasure time making compiler to add needed casts automatically => You wouldn't retrieve directly the Integer value, but Any. annoying..