The general two higgs doublet model can neither give masses to neutrinos without considering a see-saw mechanism nor it can unify gravity with other three forces. To explain dark matter also we have to introduce extra symmetry to the theory. Then why do we need this theory?
The main motivations are (see e.g., 1106.0034):
- Two Higgs doublets are required in supersymmetric models as the superpotential must be holomorphic and to avoid triangle anomalies (see e.g., 9709356).
- Two Higgs doublets could introduce extra sources of CP-violation, an ingredient for baryogenesis (see Sahkarov's conditions). The CP-violation could be explicit or in fact spontaneous in some more exotic models.
- Two Higgs doublets could have a richer vacuum structure, with more chances for a strongly first-order phase transition. Such a transition could provide a departure from equilibrium, another ingredient for baryogenesis.
- It is a simple extension of the SM that doesn't shift the the $\rho$-parameter.