This is called bud blast, often with no damage to the plant other than a lack of bloom in the spring. With snow cover, this can appear as little or no bloom above the snow line, with more normal bloom below the snow line. Injury to the plant itself is called bark split, The outer layer of bark splits away from the stem just above the ground. The damage shows up the following spring, and can kill the plant if the stem is girdled. Plants with bark split have been saved by applying melted paraffin or grafting wax over the split areas.
Roots are much more sensitive to cold than the plant stems or buds.
In the ground, the soil insulates the roots against air temperatures below freezing. The roots of potted plants need similar insulation against cold temperatures. This can be provided by a cool greenhouse, a cold frame, or by placing them closely together and covering the pots with a heavy layer of mulch. Heat damage is less obvious than cold damage. It typically shows up as a plant that simply does not thrive, and instead slowly declines and dies over a period of years.