Unfortunately, many of the longest reigning dynasties have an unverifiable date of foundation due to poor record keeping and the influence of foundation myths. If we're excluding cadet branches and tracking a single line's rule, the two primary contenders for the longest reigning dynasty are the Japanese Yamato Dynasty and the Ethiopian Solomonic Dynasty.

The Japanese imperial family claims to have ruled from approximately 660BC, when it was founded by a probably mythical emperor named Jimmu. The earliest emperor considered to have a plausible but unconfirmed historicity is Emperor Sujin who, according to Japanese legend, began his rule in 98BC. It is not until Emperor Kinmei, traditionally regarded as the 29th Emperor of Japan, that we have enough evidence to verify probable dates for his reign (beginning in 539 and ending in 571). As you noted, the Yamato Dynasty still survives and nominally rules Japan today.

The Solomonic Dynasty of Ethiopia claims to descend from Solomon of Israel, who is traditionally believed to have reigned sometime in the mid-9th century BC. Similar to the early Japanese emperors, both the foundation of the dynasty and the early rulers are unverifiable. Unfortunately this is even more difficult for the Solomonic Dynasty, as many records from early Ethiopian history were supposedly destroyed in the late-9th century AD during the (difficult to verify) reign of Gudit, who according to tradition destroyed Christian monasteries in favor of Judaism. The earliest verified ruler we have for the dynasty is Yekumo Amlak, who "restored" the dynasty after several centuries out of power. He claimed to have traced his ancestry to Dil Na'od, the last monarch of Aksum, who himself comes from a lineage that is very difficult to verify due to a lack of evidence for its first few rulers. Traditionally the Kingdom of Aksum was founded by Zoskales sometime in the first century AD, and there is limited contemporary literary evidence for his existence. His successor, Gadarat, is mentioned with his son and successor, Begyat, in inscriptions and artifacts found in Yemen and the Horn of Africa, so the record becomes a bit clearer around 200 AD. Once you hit Endubis, who reigned sometime in the mid to late 3rd century, we have coinage and other solid archaeological evidence to verify the reigns of the rulers. All of this still, however, rests on Yekumo Amlak's claim of descent from Dil Na'od being true, which is difficult to verify.

So, it really depends on which monarchs you believe are founded enough in history to be considered. If we take the first rulers with historically verifiable dates for each dynasty, we come up with 539 - 2016 (1477 years) for the Yamato Dynasty and ~270/1270 - 1974 (1704/704 years) for the Solomonic Dynasty depending on whether or not we accept Yekumo Amlak's claimed lineage. Beyond that, it becomes far too speculative based on what we are told by myth and proto-history but cannot confidently verify.

As you can see, it gets a little messy when discussing these dynasties as record keeping was likely limited by the constraints of the time and the probability that many documents were destroyed or lost sometime in the past. It's difficult to verify claims in the history of entire kingdoms and empires at that point in time, verifying the historical claims of a family is even more so. Someone with more knowledge and experience in this particular field may be able to clarify some points or correct any mistakes I may have made, but hopefully this answer helps.

As far as we know, it is the verifiably longest reigning continuous dynasty in all of history. Traditionally, the start is considered to be Emperor Jimmu, 7th century BC. However, the first emperor we can trace the genealogical record back to is Emperor Keitai, who reigned at the start of the 6th century AD.

There may also be a case for the Tamil dynasties, Pandya, Chola, and Chera, which was referred to in Asoka's pillars around 300BC and existed until the 14th century when they were absorbed by the Vijayanagara Empire that unified the Dravidian kingdoms, though there are interregnums.

There's others of comparable length that have less historicity. The Ethiopian Solomonic dynasty claims direct patrilineal descent from King Solomon of Israel, but was only re-established in 1270.