The details of the brothers' disagreements are nowhere recorded, but their result was of historical significance, causing the old North-kingdom of the Dúnedain to be split into three realms. Amlaith's realm was reduced to the lands west of the River Baranduin and the Weather Hills, a region that came to be known as Arthedain. The eastern lands of Arnor were divided between Amlaith's brothers, creating the new kingdoms of Rhudaur to the north and Cardolan to the south.
Whatever the cause of the dissension between Amlaith and his brothers, the fact of his direct descent from Isildur was never in question. Through Amlaith, the Kings of Arthedain carried the bloodline of Elendil down through the generations.
The date of Amlaith's birth appears only in The History of Middle-earth volume XII, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
The meaning of Amlaith's name is extremely obscure. The initial am- element is probably 'up', but -laith is much more difficult to interpret. This element appears in the name Limlaith, an Elvish name for the River Limlight, so on this rather shaky basis we might translate laith as 'light'. Other sources, in a context quite unrelated to Amlaith's name, translate the ending -laith as 'spirit'. These references date back to the very earliest phase of Tolkien's legendarium, and it is unclear whether they are relevant in this case, but they suggest something like 'high spirit' as at least a possible interpretation.
Amlaith's younger brothers (whose names were frustratingly never recorded) were the founders of the kingdoms of Cardolan and Rhudaur. It should be noted that there is no specific evidence that Amlaith had only two brothers, and indeed there is the vaguest of hints in The Peoples of Middle-earth that he may have had more, but if so they left no mark on the history of Middle-earth.