We study the non-linear growth of cosmic structure in different dark energy models, using large volume N-body simulations. We consider a range of quintessence models which feature both rapidly and slowly varying dark energy equations of state, and compare the growth of structure to that in a universe with a cosmological constant. We use a four-parameter equation of state for the dark energy which accurately reproduces the quintessence dynamics over a wide range of redshifts. The adoption of a quintessence model changes the expansion history of the universe, the form of the linear theory power spectrum and can alter key observables, such as the horizon scale and the distance to last scattering. We incorporate these effects into our simulations in stages to isolate the impact of each on the growth of structure. The difference in structure formation can be explained to first order by the difference in growth factor at a given epoch; this scaling also accounts for the non-linear growth at the 15 per cent level. We find that quintessence models that are different from Λ cold dark matter (ΛCDM) both today and at high redshifts (z∼ 1000), and which feature late (z < 2), rapid transitions in the equation of state, can have identical baryonic acoustic oscillation (BAO) peak positions to those in ΛCDM. We find that these models have higher abundances of dark matter haloes at z > 0 compared to ΛCDM and so measurements of the mass function should allow us to distinguish these quintessence models from a cosmological constant. However, we find that a second class of quintessence models, whose equation of state makes an early (z > 2) rapid transition to w=−1, cannot be distinguished from ΛCDM using measurements of the mass function or the BAO, even if these models have non-negligible amounts of dark energy at early times.