Most running socks nowadays provide extra cushioning in critical areas, such as under the balls of your feet and the heels, without making the socks too thick overall. This helps ventilation and comfort levels as your feet don't get too warm and sweaty, which can reduce the chances of swelling.
This is all well, but you'll mainly benefit from wearing thin, padded-running socks when the weather is nice and warm outside (or when running on a treadmill). When the weather turns grim, you might need more thickness to keep the joints nice and warm, reducing the risk of injury. You don't necessarily need thick hiking socks, but getting a full-size, warmer running sock for the winter might help.