There are some scores that are so effective you don't notice them most or all of the time (offhand I can't recall any for obvious reasons); there are some scores that are so effective that you do (Kubrick's often). Probably the most notable composer of significant, memorable, and popular movie tunes is Henry Mancini (obvious examples). But he could also write a riveting background piece, like the opening track for Touch of Evil (which as you know was not intended by Welles to be superimposed over the visual opening).

Songwriter-directors are an interesting hyphenate--not too many of them (Chaplin and Carpenter come to mind).