you could take off the one-sided rivalries where one side cares far more than the other:
ManU v ManCity
Everton V Liverpool.
Also, a special on the EPL showed that the Mereyside derby is one of the friendliest in England is also known as the friendly derby. ManU vs. Liverpool is far more poisonous than ManU vs. manCity, or Liverpool vs. Everton and is widely regarded as the biggest rivalry in the EPL.
Aregntina vs. Brazil will always be bigger than Argentina vs. England. Just not the same, despite the war.
For me:
Club: Boca vs. River
International: England vs. Germany and Brazil vs. Argentina (tie)
Of course..that is high profile. On the lower profile, China vs. Japan is worst than anything coming out of Europe or South America. In 2004, Japan beat China in the Asian Cup final on Chinese soil. Allyuh bess check out the buildup to, and the aftermath of that game. Football come second