'A war' is a situation of fighting between countries, groups, etc (to fight a war / to go to war / nuclear war / a prisoner of war...). 'Warfare' means 'ways of fighting a war': biological / chemical / atomoic warfare conventional warfare: warfare in which nuclear weapons are not used.

Warfare refers to the general act and art of waging war. It does not refer to a particular war. We use the term "biological warfare" to refer to the use of biological toxins as a weapon of war. War is sometimes used in the same way of "warfare", but is more often used to refer to a particular armed conflict. You would refer to the "Second World War" or the "Great Patriotic War", but never the "Second World Warfare".