On a score, it indicates that the song was arranged by person X, with notation by person Y. What does a notation person do that an arranger does not?
- "Notated By" means basically that whatever person is credited with this transcribed this song (they didn't change it at all).
- "Arranged By" means that the person credited made some artistic choices when transcribing. Usually, this notation will occur when someone takes a song and writes it for an instrument that is different from the original, and this makes they'll have to change things, for example chord voicings. The other meaning would be that the arranger has changed some aspects of the song, for example keeping the melody but changing the harmony.
In your case, I believe this means Person Y transcribed the music "note for note" or as close as possible, whereas Person X arranged it, changing the piece, either due to different orchestration or creative modification.
I like your name, by the way.
Well, notation. The graphical typesetting. What an arranger does would be audible in the produced music. What a notationist does would just be audible in the cursing from the musicians deciphering the score.
But basically the notationist is not responsible for the sound but just for the arrangement of ink on paper.
'Arranged by' refers to less than composing, more than transcribing or orchestrating. An existing work provides the main musical content.
'Notation by' is not generally seen. It means whatever the guy who wrote it wants it to mean! We might guess that it's about a reluctance to use 'engraved by' or 'copied by' where actual engraving or pen-and-ink aren't involved. But it might refer to transcription - creating notation from a recording - or even to changing a guitar piece from tab to notation.
So we can only make a guess at the answer to your question. Maybe the context gives more clues? What's the score?