I'm not an expert but I think it's because of the shapes of the continents and oceans and their effect on ocean currents.
The Arctic Ocean is effectively a northern extension of the Atlantic, with relatively little water exchange with the Pacific. So, currents flow primarily north-south, allowing warm water to be exchanged.
By contrast, the Southern Ocean is not hemmed in by land, and so there's nothing stopping currents flowing longitudinally rather than latitudinally - which is exactly what happens. Cold water is exchanged for other cold water, at least on the surface, keeping the ocean and surrounding areas cooler.
(Please correct me if I'm wrong!)
There's also very little land stopping the wind around the Southern Ocean, so the cool temperatures in the Tierra del Fuego might not be your only issue!