As different as weather is from climate, these two concepts are often used interchangeably in everyday language. At the end of the day, they are both descriptions of atmospheric behaviors, aren't they? According to NASA, the key difference between weather and climate lies in the different time scales on which they operate. Weather refers to the atmospheric condition over a short window period of time (hour, day, week), whereas climate is the accumulation of weather over an extended period of time. Geographic scale can add another layer of distinction to this. Weather usually covers a relatively limited area while climate over a large region.
One of the fundamental challenges facing climate scientists and science communicators is how to effectively communicate long term changes in the atmosphere (climate change) to the public who may confuse climate with weather. As rational as we may wish to be, our minds are actually prone to many mental heuristics and biases (Tversky and Kahneman 1974). For instance, when being asked about long term changes in hurricane strength - "do you think hurricanes have become stronger?"- coastal residents are more likely to take into account their observations of characteristics of the most recent hurricane landfall such as maximum wind speed and storm surge, rather than the actual trend of hurricane strength over the recent past (Shao et al. 2016). This is classic illustration of availability bias, referring to the mental tendency to turn to the most accessible and retrievable events (Tversky and Kahneman 1974).
The seemingly clear conceptual delineation between weather and climate can be blurred if entering extreme weather events. Scientists are now taking on a big task to disentangle specific extreme weather events and climate change. Whether or not Louisiana's Epic Flood (2016) is a direct product of climate change may be still subject to debate. Scientists nevertheless can say that climate change is certainly driving up the frequency of extreme weather events, essentially making 1000-year (the chance for an event to occur in any given year is 1 in 1000) or 500-year events become 50-year or 10-year events. When people start to see the overall pattern that extreme weather events have become the new norm across the globe, hopefully it will make the case of climate change more convincing and scientists' communication less challenging.
Shao, W., Xian, S., Keim, B. D., Goidel, K., & Lin, N. (2016). Understanding perceptions of changing hurricane strength along the US Gulf coast. International Journal of Climatology.
Tversky A, Kahneman D. 1974. Judgment under uncertainty: heuristics and biases. Science 185(4157): 1124–1131.