What artists have impacted the modern world the most?

This is a challenging question, but three artists jump out: Claude Monet, Vincent Van Gogh and Pablo Picasso. All three helped move painting away from a strict realism in which the artist tried to reproduce the most accurate possible stylized imitation of objects in the real world, to art that pushed the boundaries of perception in different ways. It's important to note that all three artists worked after the camera made it possible to capture the kind of realistic images that before had been the realm of painting and sculpture in the Western world. They are influential because they changed the way people see. 

Claude Monet would have argued strongly that his approach, called impressionism, was scientific, and that he strove to capture an accurate and authentic depiction of a landscape or scene. What Monet did was to go outside and paint whatever he actually saw. He would frequently paint the same object over and over again at different times of day, in different seasons and in different weather. What he showed was that our impression of an object can be almost entirely different based on when and how we see it. For example, he famously painted haystacks at different times of year, in different weather and at various times of day. A haystack looks different in the fog or the rain than it does in the snow and different at dusk then mid-day. Sometimes my students will say: any four-year-old could have painted that haystack. But what Monet painted was what he saw, and if the weather made the haystack look like a wavy blob, that's what he recorded. What Monet did was establish the subjectivity of seeing. When we paint a haystack from memory, we paint an idealized haystack, not necessary the haystack as it appears in front of us in a particular moment: Monet helped break that way of seeing apart.

Van Gogh used vivid colors and laid paint thickly on his canvasses to achieve emotional, rather than realistic, effects. He famously said "real painters do not paint things as they are ... they paint them as they themselves feel them to be." For example, unlike Monet, he painted his famous canvas, Starry Night, from memory and used it to convey a spirit of emotional tumult in the swirling sky.

Picasso is an extraordinarily influential painter whose long life spanned two world wars. He is most famous as one of the inventors of cubism, a method of painting that broke objects into their geometrical parts and reassembled them without necessarily paying attention to creating something representational but often overlaying one point of view on top of another: thus a human being might be reduced to geometric shapes representing his front, back and side views and then all the shapes placed together. This abstract or non-representational art went far beyond Monet or Van Gogh's work in separating a painting from its subject, revolutionizing the art world in the twentieth century.