When snow falls at a heavier rate (comes down heavier), it has the ability to accumulate faster and override above-freezing temperatures.
Snow (which by default begins at below-freezing temperatures) falling through above-freezing temperatures begins to melt. This causes the air around it to cool. Effectively, this allows the snow after that to fall as snow all the way to the ground, and over time, stick.
Heavier snow will allow for this effect to occur. However, with a light snow, this process is very slow to occur. Many times light snow is at the end of a storm, or in some other situation where the snow is brief. Because of that, accumulations are much harder to come by, especially if temperatures are only at 32, and especially if they are 33 or 34.
Feel free to reach out if you have any questions or would like more of an explanation.