Meteorological is not the name of the science. It is the describing word for things that belong to that science — weather as scientists measure it: rain, wind, heat, storms, air pressure. A meteorological report, meteorological data, a meteorological office: all of these are about weather, written in a formal, scientific voice. In ordinary talk people almost never say it. They say weather. 'Bad meteorological conditions' is the tidy official phrase; 'terrible weather' is what a person actually says to a friend. Though actually, it is also not about meteors (shooting stars). The two words grew from the same old idea of 'things high in the sky', but in today's English they have gone different ways, and mixing them up is a very common honest mistake. Use meteorological when the context is science, news, or an official warning. Use weather everywhere else.