I think Noxus would win.
Demacias army might be better but Noxus outnumber them greatly. We know Noxus and Demacia had some even battles in the past, but those were when Darkwill was fighting wars at every front. Now Swain, Darius and a mysterious third person are in power, and that would make a huge difference.
First of all, Swain is the kind of guy to always have a plan. Fighting unwinnable wars is the reason why he threw Darkwill from the throne, Swain wouldn't make the same mistakes. He is a tactician and with his demons power he is able to evaluate the war in a way nobody else can. Second, Noxus now has the Trifarian Legion, although we don't know how big it is, they should be better trained than common Demacian soldiers. And lastly, Noxus has no scruple. Chemical weapons, magic (while less effective is still a potent weapon that can kill hundreds at a time), and sending soldiers to their death only to have new ones follow them are all thing Noxus is willing to do.
Demacia would maybe win more battles, but they just don't have enough manpower to win the war.