There are a number of musical "styles" contained in the Sacred Harp -- early American music, folk hymns, psalm & hymn tunes, fuging tunes, odes & anthems, campmeeting songs, gospel songs, among others. I will record some of my attempt to sort this out in my own mind, which may or may not prove helpful to others. I rely on R. Paul Drummond's exposition and definition of Primitive Baptist song, as well as comments from Dr. Warren Steel. Much of this is left up to wide variations of opinion. What follows is a hodge-podge I've collected.
A Portion for the Singers (Drummond) identifies four broad categories of music found in Primitive Baptist hymnals: Southern Folk Hymns (folk hymns, Sacred Harp, fuging tunes, anthems); Mason-Bradbury Hymns, Gospel Songs (Kieffer/Showalter variety, Sankey/Bliss variety, Stamps/Baxter/Brumley variety), and Traditional Protestant Hymns. This is very similar to what is found in Sacred Harp, with the exception of the Stamps/Baxter/Brumley variety. Attempts to categorize music can be fraught with difficulties. For example, Drummond points to "Ortonville" and "Toplady"/"Rock of Ages" as examples of the problems of such categorizations (A Portion for the Singers, R. Paul Drummond, 1989, p. 18) -- "Ortonville" long since becoming a part of "folk tradition" and "Toplady" being a very traditional Protestant hymn. Yet, because of their origins, both are part of the Mason-Bradbury category of song. There is no reason that some categories cannot overlap, or a some be found in more than one category.
Psalm and hymn tunes (Examples: Old Hundred, Ninety-Third Psalm, Bethel) -- generally short tunes designed for one stanza of metrical hymn.
Folk hymns (Examples: Wondrous Love, New Britain, Pisgah) -- contrafactum of a secular folk song, ballad of religious experience, and camp-meeting spiritual
Campmeeting Songs (Examples: Sweet Morning, Farewell Vain World, The Morning Trumpet)
Campmeeting songs incorporate simple words with "simple tunes and meters [that] required no skill and therefore invited everyone to participate in the service." -- Bernard A. Weisberger, They Gathered at the River, Boston, 1958, p. 148
"The refrain or chorus is perhaps the predominant feature, not always connected with the subject-matter of the stanza, but rather ejaculatory. In some instances such a refrain was merely tacked on to a familiar hymn or an arrangement of one." -- Louis F. Benson, The English Hymn, New York: George H. Dornan Co., 1915, p. 293 (The interrupting refrain is a short phrase interpolated between the lines of the primary text.)
Paul Drummond sees a difference between "a first generation folk hymn" and those later composed "in the style". "Like most attempts at stylistic imitation these pieces tend to exploit clichés and most take on a rather uninspired, pedestrian quality." -- A Portion, p. 252 (In this he implicates a number of early Sacred Harp composers -- including both Rees brothers and Edmund Dumas. I would disagree with his assessment of their "pedestrian quality".)
Fuging tunes (Examples: Stratfield, Ocean, Mount Pleasant) -- usually begins with a homophonic section, in the course of which a definite cadence is reached; a new start is then made in which each individual part makes an entrance in succession, often utilizing some form of imitation; and the final phrase most often ending homophonically. (Irving Lowens and R. Paul Drummond)
Anthems (Examples: Easter Anthem, Rose of Sharon, David's Lamentation) -- "a musical setting for chorus of a non-metrical prose text, sectionalized by changes in tempo, meter, key, texture, and sonority." (John Worst)
Gospel song (Examples: Let Us Sing, Marriage in the Skies, Sweet By and By)
"Musically, the typical gospel song is in a major key, in common (4/4) time, with numerous repeated notes in a melody that is more interesting than the parts that accompany it." -- A Portion, p. 288 "Gospel songs of this type often display repetitive rhythmic and melodic features and rudimentary antiphonal and responsorial textures." -- A Portion, p. 300
"The gospel hymn was developed to meet the needs of revival and prayer meetings...The mood of the text might be optimistic or pleading; the music was tuneful and easy to grasp. The rudimentary harmonies, the use of the chorus, the varied metric schemes, and the motor rhythms were characteristic. A march-like movement as in 'Shall We Gather at the River' was especially typical. The device of letting the lower parts echo rhythmically a motive announced by the sopranos became a mannerism which was abused by later writers...The best of the gospel hymns have a direct simplicity which has appealed to singers ever since the appearance of the first gospel hymnals." (Janer places George F. Root among the earlier composers of the style.) -- Albert Christ-Janer, American Hymns Old and New, Vol. 2. New York: Columbia University Press, 1980, pp. 365-66
"The melodic range was designed for congregational song and was therefore limited to that of the untrained voice from about middle C to top-line F. Harmonies were generally primary triads, although secondary triads, borrowed minor chords, and secondary dominants became part of the harmonic vocabulary...Most characteristic of the gospel song was a contagious chorus or refrain that summed up the text's meaning in a succinct and memorable manner. -- W. K. McNeil, Encyclopedia of American Gospel Music, p. 293
The gospel song and Mason-Bradbury hymns tend to closer harmony than many of the early American tunes and folk songs. In close harmony, the alto and tenor parts tend to parallel the melody so that all three parts can be played on a keyboard. The bass part is not melodically tied to the soprano, yet fulfils a harmonic function. This describes the hymn tunes of Mason, Hastings, Bradbury, etc. as well as much early gospel music, especially of the more homophonic variety. (Warren Steel)
"The music of the 'Better Music boys' found in the hymnals of Primitive Baptists are typically short, strophic hymn-settings in major keys with limited ranges and easy tessituras." -- A Portion, p. 278 (Drummond gives "Brown" by Wm. Bradbury as an example.)
Methodist theatrical (Examples: Enfield, Dartmouth)
A few songs in the Sacred Harp are described by scholars as the "Methodist-theatrical style” of hymn-tune. These are imitations of those in 18th century British collections like Butts' Harmonia Sacra and Madan's Lock Hospital Collection. (Warren Steel) Someone described these as having "theatrical embellishments and a gallant cadence".