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.

CCl FAQs | WMO

www.wmo.int/pages/prog/wcp/ccl/faqs.php

Climate variability – The way climate fluctuates yearly above or below a long-term (monthly, early or decadal) average value. It may be result of natural, large scale features of climate like SPCZ, ITCZ, WPM, ENSO etc.

Climate change – Long-term(30-year) continuous change (increase or decrease) to average weather conditions or the range of weather.

Climate variability refers to the variations in the mean state and other statistics (such as standard deviations, the occurrence of extremes, etc.) of the climate on all spatial and temporal scales beyond that of individual weather events. It can be attributed to the natural internal interactions of the components of climate such as ENSO, NAO, PDO which are recurring in the climate and also to external forcings.

Climate change refers to a change in the state of the climate that can be identified (e.g., by using statistical tests) by changes in the mean and/or the variability of its properties, and that persists for an extended period, typically decades or longer. Trend of climatic variables over longer time period and evidences of their impact act as indicators of the climate change.

Climate variability - The way climate fluctuates yearly above or below a long-term average value.

Climate change - Long-term continuous change (increase or decrease) to average weather conditions or the range of weather.

As per IPCC definition,

Climate Climate in a narrow sense is usually defined as the "average weather," or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands of years. The classical period is 3 decades, as defined by the World Meteorological Organization (WMO). These quantities are most often surface variables such as temperature, precipitation, and wind. Climate in a wider sense is the state, including a statistical description, of the climate system.

Climate Change Climate change refers to any change in climate over time, whether due to natural variability or as a result of human activity. This usage differs from that in the United Nations Framework Convention on Climate Change (UNFCCC), which defines "climate change" as: "a change of climate which is attributed directly or indirectly to human activity that alters the composition of the global atmosphere and which is in addition to natural climate variability observed over comparable time periods."

Climate Variability Climate variability refers to variations in the mean state and other statistics (such as standard deviations, the occurrence of extremes, etc.) of the climate on all temporal and spatial scales beyond that of individual weather events. Variability may be due to natural internal processes within the climate system (internal variability), or to variations in natural or anthropogenic external forcing (external variability)

http://www.ipcc.ch/ipccreports/tar/wg2/index.php?idp=689

The Chapters 8 and 10 of the 5th IPCC report will clearly explains the climate change and climate variability

http://www.ipcc.ch/report/ar5/wg1/

For an example; If we look at the graph of All India Summer Monsoon Rainfall from 1871 to 2017 (IITM), the increase and decrease shows the change in climate during this duration. However, if we see the inter-annual variability, there are number of flood and drought events

Climate variability- Deviations in the mean state and other statistics of the climate on all temporal and spatial scales, beyond individual weather events.

Climate change- Climate change refers to a statistically significant variation in either the mean state of the climate or in its variability, persisting for an extended period (typically decades or longer).