At first we are led to believe that Mance found the Horn of Winter north of the wall and brought it south with his army to break the Wall. The horn is captured when Stannis Baratheon's troops routed the army, and was made fuel for Mance's pyre. Then we discover that..
Mance was bluffing - the horn was a fake (and not killed in that event!)
At around the same time or soon after, Euron Greyjoy turns up at the Kingsmoot with the dragon horn called Dragonbinder, claiming it came from Valyria. He gives it to his brother Victarion when sending him to 'fetch' Daenerys Targaryen (I can imagine how poorly that will go!).
This horn supposedly binds or controls dragons, but almost anyone that blows it suffers death from 'blackened lung' (though presumably Daenerys herself could blow it safely, given she has dragon's blood in her veins).
Although the Horn of Winter was supposedly somewhere north of the Wall, and Dragonbinder was sourced from Valyria, some of the answers to What was Euron "Crow's Eye" Greyjoy doing in Valyria? display skepticism that Euron was ever in Valyria (so he might have actually obtained it north of the wall).
There are also some tentative links between dragons and 'things north' in that:
- Dragons can make dragon glass which is capable of killing White Walkers
- If dragons can make short work of Harrenhal, they can also make .. a breach in a wall made of ice! It is noted in the this answer on Dragons that..
Dragons are believed to be intrinsically tied to magic and the seasons of the world. Since dragons became extinct from Westeros, the power of magic dwindled and winters grew colder.
Which in turn suggests that if Dragons were brought to the Wall and destroyed it, the problem of the others would also abate, since the others are known to be associated with the cold/winter, if not only moving around in it, then perhaps even causing it.
Of course none of these tidbits are definitive evidence that the two horns are one and the same, so my question boils down to: Is there some evidence I missed that definitively rules out that the two horns are actually a single horn with two names?