There's no real need for a formal declaration of war since the Borg don't utilize diplomatic relations like others do.
A declaration of war, once adopted, can also legally change the government inside the declaring state. How the Federation operates for the "duration." Government programs and services could be temporally restricted or suspended so as the Federation could utilize those resources (money and supplies) to protect itself. During wartime, the rules for the Federation council could change, legislation and budgetary matters could be passed much more quickly.
A formal authorization would bring into play mutual defense treaties with allies like the Klingons.
An authorization would change the way Starfleet operates as well, Borg ships could be fired upon and destroyed without warning. Or computer viruses employed. A transfer of ships, assets and resources from exploration to defense.
Civilian freedom of movement could be curtailed, areas of the Federation where Borg incursion were expected might be place off limits. Something the Federation council might legally be forbidden to do during peacetime.
So, it isn't just about sending a piece of paper to the Borg.