Weather is generally defined as the state of the atmosphere at a particular time and place. Often a description of the local atmosphere using the seven basic elements of weather: temperature, air pressure, humidity, visibility, wind speed, wind direction and sky conditions.
Weather is very specific and local and can change from one minute to the next as well as from one community to the next. This is noticed easily and often as one town experiences rain while a nearby town receives sunny weather. Knowing how weather can change rapidly explains the difficulty with predicting it, especially when attempting to predict farther than a day or two away.
Climate, on the other hand, is much more general and describes regions of weather over a longer period of time throughout a larger geographic area. Average weather is a good way to think of climate. While Baltimore and D.C. can be having drastically different weather, no one talks about the different meteorologic climates of the two cities.
Many people tend to head towards the equator for vacations because they know of the warmer, sunnier climates there which last many months, although locally somewhere may be experiencing varying weather conditions lasting much shorter time periods.