When you write out a melody this way it shows where the notes are found within the (universal) major scale, making it easier to play it in any key on your instrument. It also allows you to notice patterns or commonalities that you might not otherwise notice when you segregate tunes by key.
For example, I noticed the occurrence of a sharpened 5th note in several of the Caribbean melodies I've been learning, especially those with a minorish sound. This may be an indication of a dominant 3rd chord which creates tension that is ultimately released by the 6 chord (a minor chord), in much the same way that the naturally dominant chord (the 5 chord) creates tension that is then resolved when it goes to the 1 chord. In other words, 5 is to 1 as 3 is to 6.
To provide an example of this numeric notating I have chosen Old Joe Clark because it is both simple enough and weird enough to be good fodder for analysis. Old Joe Clark is what old-timers call a "modal" tune, which basically means that its tonal center is based on a note other than note 1 of the major scale. However, the notes of the major scale are still 100% present in Old Joe Clark...it just places more emphasis on notes 2 and notes 5 of the major scale than note 1, as you can see in the numeric transcription below.
|Old Joe Clark numeric transcription|
The brilliant thing about notating the melody in this way is that it makes all keys relative, so with a little practice you could just as easily play Old Joe Clark in any of the 12 keys simply by knowing where the notes of each of the major scales fall - in any position - on your instrument. I am thinking of adopting this notation method for many of the tunes I am learning. It provides significant insight into the construction of melodies.