In this paper, we define a cohomology theory for differential Lie algebras of any weight. As applications of the cohomology, we study abelian extensions and formal deformations of differential Lie algebras of any weight. Finally, we consider homotopy differential operators on ${\mathrm{L}}_{\mathrm{\infty }}$ algebras and 2-differential operators of any weight on Lie 2-algebras, and we prove that the category of 2-term ${\mathrm{L}}_{\mathrm{\infty }}$ algebras with homotopy differential operators of any weight is same as the category of Lie 2-algebras with 2-differential operators of any weight.