It is well known that the associated homotopy categories of Lurie's stable infinity categories are triangulated categories. In most cases (i.e. excluding some special cases of triangulated categories without models) this is how the triangulated structure arises.
In light of the relative simplicity of stable infinity categories compared to triangulated categories, at least in writing down the basic definitions, when is it preferable to use one over the other? My question is focused primarily on the case of (subcategories of) the derived category of complexes of sheaves on a space (or site).