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