The formal difference is great. Without declaring war you get ambiguity, and unpopular wars like Vietnam, where veterans came home to be despised and spat on by their fellow American citizens. Like my comment above says, if the people want a war then Congress has the power to declare one, and be clear about it. Without this level of assured commitment, you get war engagements like Iraq -- a country which never attacked us, and we matter-of-factly invaded.