North Korea has branded Japan's PM Shinzo Abe an "imbecile" and "political dwarf", accusing him of mislabelling its latest weapons test.
Mr Abe condemned the North for "repeated launches of ballistic missiles" after two projectiles were fired on Thursday.
But the North insisted it was testing a "super-large multiple-rocket launcher".
On Saturday, state media said Japan "may see what a real ballistic missile is in the not distant future".
North Korea is banned from firing ballistic missiles under UN Security Council resolutions.
The country is under various sets of sanctions over its missile and nuclear programmes. Lifting the sanctions has been a key aim of the North in talks with the US - Japan's ally - but these have stalled since a summit between its leader Kim Jong-un and President Donald Trump broke down in February.
What's the latest spat about?
The North fired what observers in South Korea called two "unidentified projectiles" from its South Hamgyong province into the Sea of Japan on Thursday.
Condemning the launch, Mr Abe said: "North Korea's repeated launches of ballistic missiles are a serious defiance to not only our country but also the international community."
The North issued photographs said to be of Mr Kim inspecting the launch.
North Korea's KCNA state media said on Saturday: "It can be said that Abe is the only one idiot in the world and the most stupid man ever known in history as he fails to distinguish a missile from a multiple launch rocket system while seeing the photo-accompanied report."
It added: "Abe may see what a real ballistic missile is in the not distant future and under his nose. Abe is none other than a perfect imbecile and a political dwarf."
Vipin Narang, associate professor of political science at the Massachusetts Institute of Technology (MIT), said neither side was necessarily right.
The projectile is believed to have been a KN-25, a rocket many analysts consider to be a "ballistic missile" due to its large size, and because it follows an unpowered ballistic trajectory for part of its flight.
"The North Koreans are playing a semantics game, basically, and by the same token so is Abe," said the MIT specialist.
"When he said ballistic missile he's clearly trying to signify these tests are a violation of UN Security Council resolutions," he told the BBC.
"North Korea is saying ... it's a multi-launch rocket system, which is different from a ballistic missile."
But the bigger story, the security expert added, is that North Korea could be preparing for longer-range ballistic missile launches.
"The bigger line... is that Abe may soon see a ballistic missile under his nose. It seems to suggest that they are gearing up to start testing longer-range systems," he said.
Where are we with talks?
Negotiations between North Korea and the US remain stalled since the collapse of February's summit in Hanoi.
Mr Trump and Mr Kim did meet again in the Demilitarized Zone that divides the Koreas in June and agreed to restart working-level talks.
These began in October but failed to make any progress.
The North has demanded the US change its approach by the end of the year, and was lukewarm in response to a tweet by Mr Trump hinting at another meeting with Mr Kim.
In May, Mr Abe said he was ready to meet Mr Kim "without conditions", raising hopes of renewed negotiations on the nuclear issue as well as on the lingering historical issue of the North's abduction of Japanese citizens.
The Japanese were kidnapped by the North in the 1970s and 80s to help train its spies. Japan believes 17 citizens were abducted, only five of whom have since been repatriated.
But Mr Abe's offer has not come to fruition. The North said this month that the Japanese leader would never set foot in Pyongyang after he condemned an earlier weapons test.