Precision measurements of neutrino oscillation parameters have provided a tremendous boost to the search for sub-leading effects due to several beyond the Standard Model scenarios in neutrino oscillation experiments. Among these, two of the well-studied scenarios are Lorentz violation (LV) and non-standard interactions (NSI), both of which can affect neutrino oscillations significantly. We point out that, at a long-baseline experiment where the neutrino oscillation probabilities can be well-approximated by using the line-averaged constant matter density, the effects of these two scenarios can mimic each other. This would allow the limits obtained at such an experiment on one of the above scenarios to be directly translated to the limits on the other scenario. However, for the same reason, it would be difficult to distinguish between LV and NSI at a long-baseline experiment. We show that the observations of atmospheric neutrinos, which travel a wide range of baselines and may encounter sharp density changes at the core-mantle boundary, can break this degeneracy. We observe that identifying neutrinos and antineutrinos separately, as can be done at INO-ICAL, can enhance the capability of atmospheric neutrino experiments to discriminate between these two new-physics scenarios.