You could look at this from many angles, but to get started--

From a consumer culture/standard of living angle: Cotton cloth was the first textile to be made at an industrial scale. Large cotton spinning factories were built in Britain in the 1740s and in America in the 1790s. Weaving factories came soon after, in the 1780s in Britain and in 1810s in the US. These factories made tremendous amounts of fabric quickly and cheaply, which in turn meant that cotton fabric was much more available (and affordable) than ever before. I'm sure that had a big impact on style, but someone else will have to chime in here.

From a labor history angle: These cotton mills were also some of the first large concentrations of wage laborers, which started a long series of political/power/economic struggles and negotiations. Conditions were often dangerous and unhealthy and the pay was low. Many of the problems that the labor movement would eventually combat began with the cotton mills. Labor strikes and a working class self consciousness also began in the cotton mills.

In the Southern US, the growth of the world cotton industry had complex and problematic consequences. Most of the cotton used in Northern US and British factories was grown on slave plantations in the US South. Some argue that slavery in the US was dying out before the market for cotton took off. Whatever the case, slave grown cotton quickly became a major foundation for the southern economy, which had dire consequences as the 19th century proceeded, increasing the tensions that would eventually lead to the Civil War.

From a world economy angle: Cotton was one of the US's major exports (and one of Britain's major imports), and was a major component of transatlantic trade. The price and availability of cotton had a big influence on the state of the international economy. There were many causes of the Panic of 1837, but a sharp drop in cotton prices was an important trigger.