Music written on the page is the same for any style of music. The notation is the same, but how the music is performed isn’t the same. In opera and classical music, singers sing exactly what’s on the page. They memorize all the words, notes, musical directions, and markings.
That’s a lot to memorize and to get exactly right. Classical singers have to be good musicians, because they have to follow the road map exactly as the composer intended.
In musical theater, the singer follows what’s on the page, especially for earlier, more traditional songs. In later musical theater songs (material influenced by pop and rock), a singer may sing the basic notation on the page but take liberties with the rhythm and timing in the measures.
Singers often back phrase, or take liberties with the rhythms and timing while the piano continues playing what’s on the page. The singer and pianist end up back together in the next measure.
For pop, rock, R&B, and jazz, the singer takes a lot of liberties with both the melody and the rhythms written on the page. The R&B singer adds notes to the melody, commonly called riffs. Riffs are musical improvisation on the melody written on the page. The pop singer also riffs, but not as much as the R&B singer.
The phrasing in these styles of music is more causal than in classical music, and it’s more common for the singer to not hold out notes at the end of phrases, even though a long note may be written on the page. The jazz singer improvises with the pianist. The pianist may improvise a segment of the music, and the singer joins in when the piano finishes.
The singer may also improv on the melody while the pianist follows along. Jazz singers need a good ear for music so they know when to come back in based on the chord progression they hear.