Weather is about specific meteorological conditions that are experienced at given moments. For example, it is sunny, 51°F(23°C). Or it is snowing with 19mph(30km/h) winds. Weather forecasts are about trying to predict exact meteorological conditions for a future and specific dates.
Climate is about general weather conditions within a specific period of time and location. For example, September usually has 2.4"(61mm) of rain in Rome
with average 58°F(27°C) temperatures. Although it could be useful for weather forecasting, climatology is more general and spans over longer periods of time. See also Rome's climate >
Also, climate is also about the analysis of a region's geographical properties that affects temperatures, winds, precipitations and humidity. For example, the Gulf stream brings to Europe warmer winter temperatures at a certain latitude compared to the East coast of North America. Or, there are three different climate types in Ecuador because of the Andes Mountain Range. The west coast, the central mountains and the east amazon all have different climatological properties for different parts of the year.