This question is one I've never been satisfied with the answers to. That probably means that there's some good science to be done coming up with a solid answer to is. Probably the best description, though not a definition, is from Robert Heinlein: "Climate is what you expect. Weather is what you get." What is missing from the description is how to decide what to expect.
Some of it, though, works very well as a practical matter as well. If you measure a temperature somewhere (my back yard at 12:30 PM), you have observed weather -- what we got at the instant, in the exact location, that we took the measurement. This is useful to know in its own right since what we experience directly as we live our lives is weather. If my backyard thermometer showed 95 F and humid, I know that it's a good time to read a book instead of do major weeding.
If I take readings of my back yard temperature for every July 23rd for the last bunch (project: how many? Why that many and not half as many or twice as many?) of years, and average them, then I have a notion of the expectation value (a formal process from statistics which you can study to answer my questions of how many years to average). This gives me an estimate of the temperature to expect -- the July 23rd temperature climate for my yard.
Weather is made of quite a few different things beyond temperature, so climate must be as well. Beyond temperature, we have expectations about the wind speed and direction (usually pretty mild around me, and mostly from the west), how often rain falls and how much falls when it does, how cloudy it will be, how often there is lightning, how deep the ground freezes in winter, ...
Given this, if you see someone say that a weather event -- a single storm somewhere, a cold temperature in some modest area (and compared to the globe, the US is a modest area) has proven or disproven something about climate, you know they're wrong. It's also why you'll see many scientists talking about climate change rather than global warming. Temperature is only one part of the climate system. Temperature, on the other hand, is one of the easiest things to measure, so many climate studies do focus on that. The science problem, though, is still the whole climate system.