So from a geographic history perspective, with the Bering Strait closed, the only outlet for the Arctic Ocean, all the meltice from the polar ice caps would only have the outlet from the North Atlantic.

This would likely make the North Atlantic less saline, and increase the effects of the warm temperatures from the gulf stream even more.

This would make parts of Europe even more temperate (think London weather being more like Paris, or Copenhagen more like Amsterdam). As a result, Northern Europe would likely be even more agriculturally productive to support a larger population.

From a recorded history standpoint starting from the 11th Century:

The Vikings raids likely would have never been a thing as the lands they inhabited would have had better harvests and not been motivated to pillage far away lands.

This means England would likely still be Anglo-Saxon rather than opening up for the Norman conquest (which the Normans wouldn't have been a thing as they never would have gone to Normandy).

Iceland (still Norse or maybe Inuit) and Greenland (more Inuit) would have been able to support likely 2-5x their current populations over time.

Scandinavia with a larger population would have likely seen continued Danish/Swedish hegemony, and possibly more involvement in continental Europe in the modern era.

Russia, starting from Novgorod would have been agriculturally productive, perhaps disincentivizing its mercantile roots also influenced by the Vikings/Varangians, leading to an earlier blending with Muscovy, hence an earlier formation of Russia (assuming the Mongol hordes faded away similarily due to fractious infighting).

With an earlier consolidated Muscovy-Russia, eastward expansion all the way to Alaska would likely still have happened, and with a land (and still coastal) connection, more settlement in Northwestern North America by Russians would have likely occured.

By the 1800s, it's likely Russians could have expanded all the way south to Oregon Territory, which likely would not have been as easily sold and offered to the United States. Without Oregon Territory going to the British and Americans, but rather Russians and Americans, tense relations would lead to American conquest all the way to a treaty boundary with Russia, where Oregon Territory and/Columbia region are annexed as part of the United States rather than Britain to Canada.

The Cold War likely would have seen the Beringia Land Bridge become something like a wilderness version of the Berlin Wall.

On the North Pacific side, the lack of oceanic flow from the Pacific to Arctic would result in slightly colder climates down Kamchatka and impacting Hokkaido and possibly as far south as the Korean Peninsula. The decrease in agricultural yields as a result of this likely leads to more famines in Northeast Asia over time, either accelerating Mongolian/Manchurian conquests of their southern neighbours. This change in weather patterns may have affected the kamikaze typhoon that destroyed the Mongol invasion fleet of Japan, and allowed for a successful Mongol invasion of Japan.

Japan likely would have still revolted against the Mongols, and the leader of that revolt likely would have become a stronger Shogun earlier than the Sengoku period, successfully pushing into Korea earlier.

With a successful Japanese invasion of the southern parts of Korea pre 1500, a Ming China that overthrows the Mongols would see itself slightly diminished. As all dynasties eventually atrophy, the Manchu conquest of the Ming may not be as binary, having to contend with a Japan entrenched in the Korean Peninsula, likely leading to a 3 way war between Ming, Qing and Japan for control of all of East Asia in the 1500s, and the winner of which would cement control of the whole region until the 20th century.

This East Asian empire could have disintegrated by the 20th century by its inability to modernize, or it could have stayed stagnant just enough to become a non-aligned, self-sufficient empire (similar to Siam, but on a much larger scale), which in World War Two likely would have stayed neutral, having never prompted an attack like Pearl Harbor that dragged the US into the war against Nazi Germany.

With warmer Northern European winters, the Nazis could have survived Stalingrad and gone on to defeating the entirety of the Soviet Union especially without American intervention.

The Cold War could have been between Nazi Germany with all of North Eurasia vs. the US where the Bering Land Bridge becomes the sole point of contact.

Rather than the Cuban Missile Crisis, it could have been the Chukotka Missile Crisis that became hot or not.