Without the strong influence of humanism in 15th century Italy, we'd probably have little to make note of, regarding the contrast of artistic production of that time, to the previous centuries. Because of the humanist movement, art began to depict human beings as being important, in and of themselves. So while religious, namely christian, themes continued in art, many artists began to focus more on the actual human subjects and the space they occupied. It might even be said that, while Byznatine art had used people as mere props to display a message for the church, the art of 15th century Italy often used religious themes as a mere stage to display people (my personal observation). Other themes were explored, however, most notably, those of Greek mythology as well as purely natural themes such as landscapes.
The extremely wealthy Medici family had a huge influence in promoting humanism in artistic production as they, in affect, replaced the church as the largest patron of art in Italy, due to their wealth and love of art. Part of this influence included the depiction of mythological themes, a strong focus on the natural beauty of human form and, indeed, of all natural things.
While the influence of science may not be immediately obvious when viewing a work from this time, the influence is certainly there. A new understanding of perspective, such as three point perspective, gave artists an ability to produce a greater illusion of depth in their paintings. Advancements in materials, such as with the invention of oil paint, improved the image quality of paintings and expanded their potential for beauty and depth. It is safe to suggest, however, that the most significant product of science/technology of the time was that of the printing press. Its invention, in 1440, helped to spread humanistic philosophies and a renewed interest in science and classical standards of beauty and excellence.
There is definitely a relationship between science, humanism, and art in modern times. The technology produced by science is probably the most important mark of the twentieth and twenty-first centuries. Regarding art, it (technology) has produced new visual mediums that allow for new forms of visual expression such as animation, not to mention film. Also, new materials, such as plastics and other synthetic materials, have added new possibilities and inspiration to painting, sculptural work and architecture/interior design. For example, the unique, adaptable properties of acrylic paint, a relatively new invention, have certainly contributed to new approaches and styles in painting. But let's not forget about the computer. Arguably, the the most significant influence that technology has in art of today lies with the computer and digtal mediums. Most importantly, though, is the ability the computer provides for common people to publish artwork for potentially anyone in the developed world to see via the internet. Last year, I sold an oil painting to a woman whom I would have never otherwise met after she happened upon some images of my work on "myspace." She lives hundreds of miles away from me and would never have known that I or my art existed without this technology. This incredibly open access to the art market creates a new and exciting climate for artists and art collectors, and I believe that this degree of interaction is directly shaping the face of art at this very moment, in a way only possible as of the past 10 years or so.
Humanism has defintiely proven to be a lasting and current influence in art. I'd say modern art, as a defined movement, took (or at least attempted to take) humanism a step further in art by focusing even more intensely on the range of "feelings" that make us human. Where the focus had orignally been on depicting a human subject(s) in relation to its surroundings, it is now, in the case of expressionism and fauvism, for example, focused on depicting the inner feelings of the artist, almost completely disregarding the outside world. In this sense, humanism and naturalism have been broken cleanly apart and put, to some degree, at odds with eachother, quite contrary to what occured during 15th century Italy. Of course, humanism, in the classical sense, is still seen in art today. For that matter, so is Byzantine art and so is primitive art. What's really special about the time we live in is that there is no real style or philosophy that reigns supreme (depending on who you ask, anyway). The variety we see in art today is a reflection of the variety of people in the world. Ultimately, this is what humanism is all about and this is why I feel humanism has a greater representation in modern day art (not "modern art" necessarily) than it has ever had in history.