Dear Aliciapc,
You provide two phrases: ‘The break-out of war’; and ‘the outbreak of war’. You speak as if these two phrases were words describing the same event and were similar in grammatical structures. They are not.
One is a verb: ‘To break out’. The other is a noun: ‘The outbreak’.
So this answers your question: “Is there no difference between the two terms?” “Yes, there is a difference between the two terms --- one is a verb and the other is a noun”. Clearly, they are not interchangeable. One can not be used in the place of the other. The following two sentences carry the same information, but they are structured differently.
“I am waiting for war to break out!” “I am waiting for the outbreak of war”.
Your other question is: “What do you say?” The "break-out" of a war or the "outbreak" of a war?
The answer is: “The outbreak of war”.
“Break out” is a verb and would not normally be preceded by the Definite Article, ‘the’.
The Oxford Dictionary of English
BREAK OUT (Please note that there is no ‘the’, nor reference to a noun LD) 1 (of war, fighting or other similarly undesirable things) start suddenly: forest fires have broken out across Indonesia n(of a physical discomfort) suddenly manifest itself: prickles of sweat had broken out along her backbone 2 escape: a prisoner broke out of his cell. (All of these examples are verbs, LD)
OUTBREAK noun. A sudden occurrence of something unwelcome, such as war or disease: the outbreak of World War II.
I cannot speak for the rest of the English-speaking world, but in the United Kingdom, as the dictionary above clearly shows, the accepted phrase is: “The outbreak of war”.
I hope that this is of help to you.
Les
