It's not clear how this distinction came about, because for most of their careers the two words have been practically interchangeable. Menace comes from minari, which my Latin dictionary defines as "to threaten", while the Oxford English Dictionary defines to threaten as "to try to influence (a person) by menaces". We seem to be going round in circles. The OED has a quote from Reginald Scot, the Elizabethan cham-pion of women accused of witchcraft, about people who "stand in more awe of the menaces of a witch than of all the threatenings pronounced by God", plainly an early case of what Fowler called Elegant Variation.
No one decided that a menace should refer to the agent of danger while a threat was the danger itself - that a menacing cloud, say, should bring the threat of rain - it just happened. Nor is it obvious why there should be something far more disturbing about a menace than there is about a threat: one never hears talk of idle menaces. Perhaps it's just another example of the Latin-derived synonym carrying more weight than the Germanic one (terror, for example, is more alarming than fear), which has always made me suspicious of those Guides to Good Writing that tell you to choose the Anglo-Saxon word in preference to the Latin one. Naturally, like all strong words, menace is vulnerable to over-use, but not even the puerile antics of Dennis the Menace have cured one of that uneasy feeling in the pit of the stomach.