Webster's:A war is a conflict carried on by force of arms as between nations.
A Joint Resolution is NOT the same thing as a Declaration of War, either semantically or constitutionally.
What Congress did after Pearl Harbor was a Declaration of War. Our congresscritters haven't had the balls to make a Declaration since.
Yes it is.
Quite correct. Hasn't been done since World War II.
What would be the advantages and disadvantages of having Congress formally declare war (politically, etc.)?
Section. 8. Clause 11: The Congress shall have Power To declare War, grant Letters of Marque and Reprisal, and make Rules concerning Captures on Land and Water.
Declaring war is straight forward. Congress will never do this again, since it make the President pretty much in charge of everything. And your right, they will never give up the power.
Granting letters of marque. This is essentially a Congressionally approved hit on someone. They do it every day and twice on Sundays.
Granting letters of reprisal. Today this is usually done via JOINT RESOLUTION.