A Declaration of War is the way to unite the country. It worked in WWII. Without such a declaration, there are the divisions we had in Korea, in Vietnam and now in Iraq. The preparers of the Constitution appear to have been very aware of the difficulties of the President maintaining a private army to do as he wished. A President calling something a 'police action' or 'carrying out UN resolutions' to avoid a Congressional declaration doesn't cut it. Why not declare war?
Because a declaration of war as defined by international treaty (which Constitutionally we are required to honor if we agreed to it) ties our hands. It limits us to specific states and specific actors. An 'Authorization to Use Force' against violations of the laws of nations is Constitutionally sound (Article 1, Section 8, Subsection 10) but doesn't tie our hands in fighting a specific country, but instead, going against a movement that is not tied to a specific nationality and doesn't fight under a single flag.
..and just to expand on the previous comment, even Paul himself found the 'authorization to use force' instead of a declaration of war completely fine when it came to Afghanistan; he voted for that.