At its core neoplatonism simply is the rediscovery and reinterpretation of the works of the ancient Greek philosopher Plato. Plato, in his writings, distinguished between a changeless and eternal realm (spiritual, heaven) and the temporary changing world we experience (earth). People began to believe that through bettering themselves they would become closer to this eternal realm. With this mind set people like Marsilio Ficino (a humanist who was the first to translate the works of Plato during the Renaissance) persuaded the wealthy merchant prince Cosimo de Medici to start the Florentine Academy and other began to sponsor great works of art and to restore ideas of classics. These ideas did include cultural aspects such as art (nudity and over exaggerating muscles was common and can be found even in painting of babies), literature (writing became more common and less religious based as more of the upper class began to read), and music (people began to focus less on religion). Finally Italian humanism began due to it's separation from Europe due to mountain ranges which prevented the black plague and invasion. Italy was divided into city states and society was less based on religion and more based on economics. With all the money merchants were making due to ports and harbors they looked for a way to help other and themselves by boosting their image. People began to idealize the "Renaissance man" someone who was educated in the classics and that spread throughout Europe.

This is a huge topic, I am only familiar with how it affected the Scientific Revolution. My source is The Sleepwalkers from Arthur Koestler.

Basically, they really liked geometry and symmetry.

Copernicus basically tried to out-Plato Plato: if the Sun represents the Form of Good, it should be the center of the universe. Besides, he really wanted an elegant geometrical model of perfect circles, not complicated epycycles - and that is very Platonist too. The circle is the perfect shape etc. etc. http://bertie.ccsu.edu/naturesci/Cosmology/Cosmo2Copernicus.html

Keplers earlier model of the solar system was based on the Five Platonic Solids, because, well, they are kinda neat. http://en.wikipedia.org/wiki/Platonic_solid#History

It is weird how little the Scientific Revolution (until Newton) had to do with modern empiricist attitudes. It was largely about looking for methods to make a more Platonic model of the solar system - more symmetrical, more geometrical, prettier. And the useful inventions were byproducts.