If anything, the ancient Greeks could be said to be more knowledgeable than modern Americans. The best and brightest Greeks received education in mathematics, geometry, rhetoric, logic, and philosophy. They defined the classical education and the liberal arts as we know them. They taught their young how to think. In contrast, Americans tend to be schooled in more technical fields and less in how to think critically.
In regards to the sum of all human knowledge, we as a species know more than ever before. Still, as far as the average person goes, it's hard to say. Your average person gets an education on liberal arts, geometry, algebra and the sorts. People are superstitious but so were the ancient Greeks.
While the times have changed and things have progressed on many levels with regard to knowledge, the Greeks took more time and placed a higher expectation on classical education. In terms of a basis of knowledge, the Greeks have a stronger foundation while the current American knowledge is based on media, mixed with social and homogenized education.
The world of ancient Greek was advanced for it's time, but I would not say it is comparable to modern American knowledge. Advances were happening quickly, bettering the lives of the average citizen. But the rate in which new technology comes out today is far greater than it was back then.