Yes, there is a difference. The former is a method which takes no arguments and returns a function of type Function1[Int, String]. The latter is a method taking a String and returning an Int.
In Scala, methods with arity-0 can be declared and invoked without parenthesis, so invoking myFunction(1) and myFunction2(1) looks the same.
If we'd convert both methods to functions, you'd see the difference in the fact that the former would take the shape of Function0[Function1[Int, String]], while the latter would be Function1[Int, String]:
myFunc: Int => String
myFunc2: (age: Int)String
scala> myFunc _
res6: () => Int => String = <function0>
scala> myFunc2 _
res7: Int => String = <function1>