I tend to notice either the melody - usually only if the singer is awesome, say Halford, Dio, and similar - or the bassline - as in, I start humming that, not the melody, and I will recognise the song by its bassline the following time I hear it.
It usually takes me a few listens before I manage to appreciate the song in its entirety.
I have known drummers describe a song only through its rhythm, and heavy metal guitarists describe a song via a succession of memorable riffs.
I reckon that most musicians apply some sort of "filter", i.e. they concentrate their attention on their main instrument while listening to a new song. I know that, as a musician, I now do just that to a far higher degree than I used to when I was only a keen listener.