The main difference between MM II (Modigliani Miller Model with Corporate Taxes) and Miller Model with Corporate and Personal Taxes is:

MM II concludes that a capital structure with 100% debt is optimal but the Miller Model states that a capital structure with 100% equity is optimal.

MM II concludes that a capital structure with 100% equity is optimal but the Miller Model states that a capital structure with 100% debt is optimal.

Both conclude that a levered firm's value will be higher than an unlevered's firm but the size of that advantage is bigger in MM II's model.

Both conclude that a levered firm's value will be higher than an unlevered's firm but the size of that advantage is smaller in MM II's model.

There is no difference between these two models, they both conclude that capital structure is irrelevant but they base their conclusion on different arguments.