The experience is very similar to touching a hot object, but are distinct biological processes.
The burning and injuries that can occur when we touch certain objects that are at very low temperatures are similar, but not identical, to those we experience when we touch hot objects. However, even though the sensations are very similar processes that produce the biological damage are different. In case of cold objects, the heat transfer from the hot object (our fingers) to cold (ice) may result freezing the water contained in the cells, formation of ice crystals, if they grow enough, can tear the cell membranes, causing the liquid contained therein to spill. The end result is the destruction of frozen tissues.
As in the case of high temperature burns, cold object must have certain characteristics to inflict damage. Thus, it should have a low temperature and a high thermal conductivity, so that the rate of heat loss greater than the heat generation in the living tissue (our fingers). Furthermore, the object must have a large capacity to store heat transferred (large mass and high heat capacity / specific heat) and the contact time should be long enough (the thermal gradient is usually relatively small compared with that of burns high temperature).Only if these conditions may cause permanent damage to the tissues.