I've been reading into this a little and there seems to be two stories. One that Japan had no cannon, or that Japan had poor cannon; in both versions they preferred close range combat with hand held firearms and boarding parties. so which is it?
Secondly if it is the case the Japanese cannons were inferior, in what way were the Korean weapons more advanced?
It's not just a matter of cannon but also the naval architecture to mount them. The Korean Panokseon was able to support heavy cannon that outranged anything the Japanese could carry. Japanese boarding tactics couldn't use such a slow ship; they relied on lighter and faster vessels to catch the enemy on the open sea, and they only used muskets and IIRC small deck guns at short range against exposed personnel.
The Japanese also just didn't have much of a navy before the war; they were preoccupied with land battles amongst themselves. The Koreans on the other hand had developed naval artillery tactics to combat pirates, but were relatively weak on land.
As to the second question I don't know every variety of artillery that the Japanese had at the time, but one interesting point is that the Koreans used the same artillery on both land and sea, so they got some dual-purposing from their investment. Their ships mounted the largest cannon they had available (the cheonja-chongtong, with a range of around 500m).
By the time of Japan's invasion of Korea, Japanese navies had roughly 3 classes of ships. The largest Japanese ships, the Atakebune (安宅船) with a displacement of 75 to 150 tonnes (assuming I'm doing my unit conversion right), had cannons in the late Sengoku. Based on ordnance orders and Luis Frois' descriptions of Nobunaga's ships, they were outfitted with countless large arquebus and 3 cannons. The smallest Kohaya (30t~) and the medium Sekibune/Hayabune (45~60t?) probably had none. The Atakebune and the Sekibune were built kind of like floating castles, with straight walls and arrow/arquebus slits.
The cannons fitted on the Atakebune were probably (it's assumption because sources don't tell us) the breech-loading swivel gun (called フランキ砲 "Frankish gun" in Japanese) that was popular all over Asia, including Japan (in as far as cannons could be, they weren't popular in Japan). They probably had a point-blank range of 300400m and a maximum range of 8001000m. Since the Atakebune, being the largest class of ships, carried only 3 cannons (at least that's what we're told how many Nobunaga's had), I think it's fair to assume the Skibune and Kohaya carried none.
Size-wise Atakebune seem to be comparable to the Panokseon. I'm not convinced Korean ships were necessarily taller or had better cannons just because 1) I am unable to find a height dimension for either ship, 2) I can't find a primary source or test data on ranges and there are a lot of conflicting numbers floating around, and 3) Korean ships greatly relied on similar class cannons, and even the swivel gun itself.
However just by counting gunslots on drawings and models, the Panokseon should be able to easily mount 20~40 cannons. So even without better cannons Korean ships should have quite easily outgunned their Japanese counterpart.