I have two dictionaries (one American English, the other British English) and they differ in writing the word “northeast”. Does this mean that in American English you should use “northeast” and in British English “north-east”? Or doesn’t it matter?
Apparently, in American English the two words indeed merge completely into one compound word "northeast." The most obvious example is how "Northeastern United States" is spelled officially (Wikipedia article).
In British English however, being somehow stricter, the two words don't get merged completely; they're written either with a hyphen that separates them ("north-east"), or they don't get compounded at all ("north east") -- it seems the region "North East England"'s name is spelled without a hyphen (another Wikipedia article). I hope this helped you in any manner.
On Oxford Dictionaries Online, the US English dictionary writes it as one word and has no entry for the spelling including a hyphen, while the British English dictionaries only have the hyphenated version (in the above link, click on "Definition of northeast in British & World English dictionary" at the bottom to see the difference). It seems that you are right and "northeast" is American English, while British English spells it "north-east".
However, Cambridge Dictionaries Online do not have any reference to the hyphenated variant.
Collins English Dictionary only has the version as one word, specifically including a reference to British English:
The Wikipedia article on compass directions shows "northeast" but not "north-east". The direction between northeast and north is called "north-northeast".
In conclusion, I would say that writing "northeast" should be generally accepted. I would only consider hyphenating the term if I was writing for a purely UK audience, if at all.
It is not just a matter of USA vs UK, you also have to consider whether we are looking at a noun or an adjective.
As a rule of thumb, compound nouns can be juxtaposed or left separate, while adjectives get a hyphen. Hence:
"I live in the north east" or "I live in the northeast"
The OED lists north-east only, with no northeast mentioned, but that does not equate to hyphens being a British English thing by a long way. You will find both north east and northeast in common use despite the OED.
Note that compound nouns are usually juxtaposed if there is an equivalent verb: "All engines eventually break down" but "We had a breakdown in communications", and that non -ly adverbs used before a noun get hyphenated: "A much-needed beer" but "The beer was much needed".