In humans not a lot. There are significant differences between blubber and fat. Though blubber is mostly made of fat it has very different composition to human fat. An animal with blubber also uses it in very different ways to the way a human uses fat. Fat in a human is an energy source, not insulation:
(blubber)... can comprise up to 50% of the body mass of some marine mammals during some points in their lives, and can range from 2 inches (5 cm) thick in dolphins and smaller whales, to more than 12 inches (30 cm) thick in some bigger whales, such as right and bowhead whales. However, this is not indicative of larger whales' ability to retain heat better, as the thickness of a whale's blubber does not significantly affect heat loss. More indicative of a whale's ability to retain heat is the water and lipid concentration in blubber, as water reduces heat-retaining capacities, and lipid increases them.[2]
Blubber is also rich in blood vessels (which the fat of a human isn't), these blood vessels adapt to temperature, this allows the animal(s) that have it to react to changes in temperature. So a whale, for example, can "re-route" blood to underneath the blubber when cold. In humans the blood is outside of the fat layer (on the skin). A constant supply of blood is required prevent frost bite (the skin dying).
Marine mammals have other adaptations too, the have a relatively low surface area to mass (when compared to say a human). So they can retain their heat more efficiently in their large bodies. Humans have a large surface area to body mass. We're adapted to life in much warmer climates, having evolved originally in Africa.
If it's really cold they also migrate into warmer waters, nursing mother wales do this.
A bigger factor in influencing humans in extreme cold is how their body reacts when it drops below the ideal 37.5C temperature. Below this temperature human organs begin to shut down, vital ones like the brain, heart, liver, etc. When this happens it's called Hypothermia. A body temperature between 32C and 35C is mild hyperthermia. Below this is life threatening.
An interesting case study is Guðlaugur Friðþórsson (an Icelandic fisherman involved in a ship wreck in the arctic ocean). He survived 6 hours in 5C water. When he got to land his core body temperature was 34C yet he showed limited signs of hypothermia. No one really understands how this process works as it's very rare but it appears some people have an ability to protect their vital organs even when their core body temperature drops below what would kill 99% of humans. Though overweight it was decided that this had a limited impact on his survival and it was his natural (unexplained) adaptations that saved him. They made a film about it The Deep/Djúpið