Overall, The Wealth of Nations was important in that it provided an intellectual basis for the growth of capitalism and free market economies. It did so in a number of related ways.
In terms of international economics, Smith's book opened the way for more free trade and less mercantilism. It started the argument for breaking down barriers to trade between nations. Those barriers are still in the process of being broken, but Smith's work was instrumental in starting the process.
In more domestic terms, this book encouraged movements towards free markets and less government intervention. Governments in those days routinely granted monopolies to various companies. Smith argued that governments should, instead allow competition between companies. He argued that the government should stay out of economic affairs.
Smith's arguments, then, were very important to the development of the modern system of capitalism in which governmental interference with trade is minimal compared to the way it was in Smith's time.