At a fundamental level, both the 21st century indie rock song and the 19th century hymn feature a very similar song structure: Verse-chorus. However, the reason for the song structures are different. Melodically, the sung verses are very similar in their stepwise motion, while the two diverge on their approach to the refrain.
Because the hymn is only an accompaniment, the structure can change depending on the proficiency of the organist playing the piece (and the musical prowess of the audience & choir). Generally, organists start out with an introduction of the piece to establish the melody and so that the congregation can recognize the hymn. Another common structural change from organists is to improvise in-between the penultimate and ultimate verses to provide emphasis and excite the congregation for the final verse. If the organist is proficient, a key change can occur in the verset between these verses.
In the case of “Jesus and Mary”, the strophic form present is very similar for an overwhelming amount of popular music: a brief introduction followed where [one of] the musical themes are stated, then a verse, a bridge into the refrain/chorus, restatement of the melody, verse two, bridge, chorus/refrain, modified verse/instrumental solo, refrain (sometimes transposed to a different key). Jesus and Mary is simply conforming to the standard structure of the time period. This is in contrast to some of the other songs on Easy Wonderful, such as “Well”, which is one extended verse that tells a story and has no strophic form.
Given that “Jesus and Mary” is not an accompaniment piece, the structure of the piece is not flexible as it is with “Rejoice, Ye Pure in Heart”. Thus, the function of the music affects it structure as well. Melodically, both “Marion” and “Jesus and Mary” are very similar. Both feature mostly stepwise motion (more information available here and here) frequently in the verses. In the refrain, however, “Marion” is still conjunct:
But “Jesus and Mary” switches to B-flat major, and outlines the B-flat major chord in the opening notes. It then stays around notes in B-flat triad throughout the chorus before returning back to G minor. See below: