For one thing, Admiral Yi may need not have bothered with defending Korea.
The fact that Japan is surrounded by deadly, deadly water has both helped them avoid invasions or forced invaders to use something they may not be great at- sailing. Such is the case of Mongol invasions.
I would wager that even a small isthmus connecting those islands the the mainland becomes the source of both profitable trade and devastating war. Japan's isolation has been important factor in history, and an isthmus would end that isolation.
I doubt Japan would have the chance to develop as it did with a mainland connection: it could be Mongol Islands.
Or maybe Japan would have succeed in taking a bite out of China and Korea, and the lands near the Sea of Japan and the Yellow Sea would all be Japan.
Hard to say more without a specific time-period.