Alkalinity refers to the capability of water to neutralize acid.
This is really an expression of buffering capacity. The presence of calcium carbonate or other compounds such as magnesium carbonate contribute carbonate ions to the buffering system. If CaCO3 actually accounts for most of the alkalinity, hardness in CaCO3 is equal to alkalinity. Conversely, unless carbonate is associated with sodium or potassium which don't contribute to hardness, soft water usually has low alkalinity and little buffering capacity. So, generally, soft water is much more susceptible to fluctuations in pH from acid rains or acid contamination.