Climate involves the whole planet. Weather applies to specific events that occur in specific places at specific times. Weather differs between countries because, first comes world climate, then comes local weather, then comes human, then come a host of other factors, then comes country!
The designation of some spaces of land as countries that lie on the surface of planet Earth may seem to be haphazard and unrelated to any natural issue, but weather conditions and climate are integral factors, along with topography and the availability of fertile land and water that moved humans to settle in one place as opposed to another.
Eventually, tribes formed into societies, then civilizations. Civilizations cement their ownership of spaces that now have fairly permanent designations as sovereign nations, each with clearly defined borders. But there have been many things beside planetary climate that have gone into the drawing of those lines.
Population grows in the parts of the planet that are most hospitable to human habitation. But with growing populations come disputes with other populations over control over water, transit from one place to another, and other natural and sociopolitical resources. Weather conditions and topography change as populations or corporations change the structure of the soil and of the land. The planet's climate, however, has rarely, if ever, been affected by the activities of humankind.
Now, it is likely that humankind has grown to such proportions that it has the potential to affect the planet's climate, and thus, local weather conditions.
Cities are built and they become heat sinks and sources of air pollution. Humans need or desire protein, and grasslands become overgrazed, the soil breaks down and desertification occurs. Humans need habitation and corporations do mining, water diversion, tree cutting and other activities that have a permanent effect on the structure of the soil, which leads to changes in weather that may seem small, but which are large enough to create the need for migration of massive numbers of humans and animals.
Extended or major volcanic eruptions created our atmosphere and thus, created many components of the world's climate.
It is becoming apparent that there are too many humans who have too much technology that has the potential to alter the planet's climate. How those alterations will affect weather, the phenomena that is specific to particular parts of the world, is up for much debate. The debate concerns whether natural sequences of climate change are the cause of changes in weather, or if humans have achieved enough influence on the planetary climate to cause changes in local weather.
It is weather, that affects the lives and livability of particular spaces on the planet that are designated as "countries". Some people fight for the best spots that have the best weather, overall, while other people have managed to eke out subsistence in places that have less optimal weather. A few hardy souls have managed to survive in places that have barely survivable weather.
Nature creates the optimal places for habitation, sustenance, survival, defense and control. Political, religious and corporate motivations and wars develop in societies to create countries in ways that are optimal for power and commerce. Originally, boundaries will go along topographical and geological lines. Geographical and sociopolitical considerations lead to other decisions as to why a national border may zig here, then zag there.
In summary, while local weather conditions are a natural and original factor in determining where humans settle, then congregate into societies, civilizations and nations, there are many other factors that go into the final setting of boundary lines of countries. And climate applies to the whole world, while weather applies to particular events that occur at specific times in specific places in the world. National boundaries are artificial constructs that are based on a host of factors, many of which do not relate to the planet's climate or to local weather conditions.