Check the respective syllabi. I think ABRSM have a prerequisite of grade 8, whereas Trinity does not. IIRC, ABRSM also has viva voce and sightreading whilst Trinity doesn't. Standard of pieces should pretty much be the same. Here's a copy and paste from the syllabi I have (might be out of date, I don't know):
Trinity, ATCL:
J S Bach Any complete sonata (from 6 Sonatas for Violin and Keyboard) Bärenreiter
J S Bach Concerto no. 2 in E major, BWV 1042 Bärenreiter
Beethoven Romance in G, op. 40 (from 2 Romances) Henle
Beethoven Sonata in D, op. 12 no. 1 Henle
Beethoven Sonata in A, op. 12 no. 2 Henle
Beethoven Sonata in F, op. 24 ‘Spring’ Henle
Bloch Nigun (no. 2 from ‘Baal Shem’ suite) Fischer
Bloch Simchas Torah (no. 3 from ‘Baal Shem’ suite) Fischer
Brahms Sonata in A major, op. 100 Henle
Brahms Sonata in G major, op. 78 Henle
De Falla Suite Populaire Espagnole Chester
Delius Légende in Eb Forsyth
Dvorªák 4 Romantic Pieces op. 75 Bärenreiter Praha
Françaix Sonatine Schott
Hindemith Sonata in Eb, op. 11 no. 1 Schott
Khachaturian Chant-Poème Boosey
Knussen Nocturne and Serenade (from Autumnal op. 14) Faber
Kroll Banjo and Fiddle Schirmer
K Leighton Metamorphoses op. 48 Novello
Messiaen Thème et Variations Leduc
Mozart Sonata in Bb, K. 454 or Sonata in A, K. 526 any reliable edition
Prokofiev Cinq Mélodies op. 35a [also known as op. 35bis] Boosey
Ravel/Catherine Pièce en forme de Habanera Leduc
Schubert Rondo in A major, D. 438 any reliable edition
Schumann 3 Fantasiestücke op. 73 Henle
Schumann Sonata no. 1 in A minor, op. 105 Peters
Smetana From My Native Country nos. 1 and 2 Peters
Stravinsky/Dushkin Chanson Russe (Russian Maiden’s Song from ‘Mavra’) Boosey
Suk 4 Pieces op. 17 (2 vols., both to be played) Simrock
Szymanowski Chant de Roxanne (from King Roger) Universal
Wieniawski Légende op. 17 Schott
ABRSM, DipABRSM:
J.S. Bach Concerto in E, BWV 1042: 1st movt (Bärenreiter BA 5190a)
Partita for solo violin no.2 in D minor, BWV 1004: r
3rd and 4th movts, Sarabanda and Giga D Bach 3 Sonatas and 3 Partitas
Partita for solo violin no.3 in E, BWV 1006: D for solo violin (Bärenreiter BA 5116)
2nd and 3rd movts, Loure and Gavotte en Rondeau R
Sonata in E minor, BWV 1023: complete. Bach 2 Sonatas for Violin and Continuo (Wiener Urtext/MDS)
Barber Concerto, Op.14: 1st movt (Schirmer/Music Sales)
Beethoven Romance in F, Op.50. Beethoven 2 Romances (Henle 324 or Henle 324/MDS)
1st movement from any one of the following Sonatas:
Op.12 no.2 in A, Op.12 no.3 in Eb, Op.23 in A minor, Op.24 in F (‘Spring’), Op.30 no.1 in A,
Op.30 no.2 in C minor, Op.30 no.3 in G. Beethoven Sonatas for Piano and Violin (2 volumes:
Henle 7/8 or Henle/MDS)
Bloch Nigun (Improvisation) (no.2 from ‘Baal Shem’ Suite) (Carl Fischer/MDS)
Simchas Torah (no.3 from ‘Baal Shem’ Suite) (Carl Fischer/MDS)
Brahms Sonata in G, Op.78: 1st movt (Wiener Urtext/MDS)
Sonata in A, Op.100: 1st movt (Wiener Urtext/MDS)
Sonata Movement (Scherzo) in C minor (Breitkopf & Härtel 6049)
Bridge Moto Perpetuo (Stainer & Bell)
Delius Sonata no.3: 3rd movt (Boosey & Hawkes/MDS)
Dvor˘ák 4 Romantic Pieces, Op.75: complete (Bärenreiter Praha H 1903)
Elgar Sonata in E minor, Op.82: 1st movt (Novello/Music Sales)
Falla Danse Espagnole (from ‘La Vida Breve’), arr. Kreisler (Schott BSS 31837/MDS)
Pantomime (from ‘El Amor Brujo’), arr. Kocha´nski. Manuel de Falla Music for Violin and Piano
(‘El Amor Brujo’) (Chester/Music Sales)
Fauré Romance in Bb, Op.28 (Hamelle/UMP)
Françaix Sonatine: complete (Schott ED 2451/MDS)
Geminiani Sonata in C minor: complete. 18th Century Violin Sonatas, Book 2 (ABRSM)
Grieg Sonata no.3 in C minor, Op.45: 1st movt (Peters EP 2414)
Haydn Concerto in C, Hob.VIIa/1: 1st movt (Henle 446 or Henle 446/MDS)
Hindemith Sonata in Eb, Op.11 no.1: any movt or movts (Schott ED 1918/MDS)
Kabalevsky Concerto in C, Op.48: 1st movt (Boosey & Hawkes/MDS)
Khachaturian Chant-Poème (Boosey & Hawkes/MDS)
Kreisler Variations on a theme of Corelli in the style of Tartini (Schott BSS 29027/MDS)
R. Kreutzer 42 Études or Caprices: no.35 in Eb and/or no.42 in D minor (Peters EP 284)
Lalo Symphonie Espagnole, Op.21: 2nd movt (Peters EP 3797a)
Leclair Sonata in D, Op.9 no.3: complete (Schott VLB 15/MDS)
Kenneth Leighton Metamorphoses, Op.48 (Novello/Music Sales)
Messiaen Thème et Variations (Leduc/UMP)
Mozart Concerto no.3 in G, K.216: 1st movt (Bärenreiter BA 4865a)
Concerto no.4 in D, K.218: 1st movt (Bärenreiter BA 4866a)
1st movement from any one of the following Sonatas: K.296 in C, K.376 in F, K.378 in Bb. Mozart
Sonatas for Piano and Violin, Vol.2 (Henle 78 or Henle 78/MDS orWiener Urtext/MDS)
Paul Patterson Luslawice Variations for solo violin (Universal 21005/MDS)
Ravel Pièce en Forme de Habanera, trans. Catherine (Leduc/UMP)
Saint-Saëns Concerto no.3 in B minor, Op.61: 1st movt (Durand/UMP)
Schumann 3 Fantasiestücke, Op.73: complete (Henle 421 or Henle 421/MDS)
Sonata no.3 in A minor WoO 27: 1st movt (Schott ED 10505/MDS)
Smetana From My Native Country: nos.1 and 2 (Peters EP 2634)
Stravinsky Chanson Russe (Russian Maiden’s Song from ‘Mavra’), arr. Dushkin (Boosey & Hawkes/MDS)
Suite Italienne: any two movts (Boosey & Hawkes/MDS)
Szymanowski Chant de Roxanne (from ‘King Roger’), arr. Kocha´nski (Universal 8694/MDS)
Tartini Sonata in G minor (‘Didone abbandonata’), Op.1 no.10: 1st and 2nd movts (Schott VLB 31/MDS)
Telemann Any one Fantasia complete from ‘12 Fantasias for Violin without Bass’ (Bärenreiter BA 2972)
Walton 2 Pieces: Canzonetta and Scherzetto (OUP)
(Do you have a teacher?)