In essence, climate variability looks at changes that occur within smaller timeframes, such as a month, a season or a year, and climate change considers changes that occur over a longer period of time, typically over decades or longer. A key difference between climate variability and change is in persistence of "anomalous” conditions - when events that used to be rare occur more frequently, or vice-versa.
In statistical terms, the curve of the frequency distribution representing the probability of specific meteorological events changes. The curve may be modified either in amplitude, shifted about a new mean, or both.
Care must be taken not to confuse variability with change. Many regions of the world experience greater variability, climatologically speaking, than do others. In some parts of the world, or in any region for certain time periods or parts of the year, the variability can be weak (i.e. there is not much difference in the conditions within that time period). In other places or time periods, the conditions can swing across a large range, from freezing to very warm, or from very wet to very dry and exhibit strong variability.
A certain amount of this is understood and accepted, instinctively, by the people in a region. What is "normal" for Denver, Colorado in terms of the frequency of precipitation events (high variability) would be "abnormal" for London, England (low variability). Thus, any single event, such as a severe tropical cyclone, cannot be attributed to human-induced climate change.
Occasionally, an event or sequence of events occurs that has never been witnessed before (or recorded before), such as the exceptional hurricane season in the Atlantic in 2005. Yet even that could be part of natural climate variability. If such a season does not recur within the next 30 years, we would look back and call it an exceptional year, but not a harbinger of change. Only a persistent series of unusual events taken in the context of regional climate parameters can suggest a potential change in climate has occurred.
The IPCC is conducting considerable efforts in trying to determine, for various hydrometeorological hazards such as tropical cyclones or tornadoes and related events such as flash floods, whether there is substantial and credible evidence of human-induced climate change. The upcoming fourth assessment report should provide more clarity on this matter.