Different people may see different connections between Plato and Aristotle’s ideas on the one hand, and modern ideas on the other. The connection I see is between people who believe things on the basis of faith and people who want to see empirical evidence for the things that they believe.
For me, Plato’s argument about the forms is similar to the way in which religious people see the world. Plato is saying that everything in the world is simply an imperfect representation of some ideal form that exists outside of our ability to perceive. The problem with this is that there is no possible way that we could know that this is true. By definition, we can never know if there are things that are beyond our ability to perceive. This is the same (in my view) as arguments about God. We cannot possibly know for certain that God exists or that he causes things to happen in our lives. However, there are many people who know that God exists because of their faith. This is their way of knowing things.
By contrast, Aristotle’s ideas are more similar to the ideas of people who believe more in science. Aristotle criticizes the Platonic philosophers because they believe things that they cannot prove. He criticizes them because they have to make up a new idea for everything that happens. In our day, scientists want to find universal rules that can be proven and that apply to every circumstance. This is the legacy of Aristotle’s ideas. By contrast, people who believe in religion over science do not feel the need to prove their rules and are happy to explain any given outcome by saying that it was the will of God. They are able to create a new explanation for each thing that happens, saying why very different outcomes all represent the will of God.
I believe that Plato and Aristotle came up with such different ideas because our world is simultaneously understandable and impossible to understand. This was particularly true in their times when we had much less in the way of scientific knowledge and technology. It is possible to see why Aristotle’s ideas came about because there are many things in our world that we can observe and understand. This is truer today when we have telescopes and electron microscopes and other such technology, but it was true even in Aristotle’s time. At the same time, it is possible to see why someone would want to believe in things that cannot be proven. There are many things in our world that cannot be proven. Why do we fall in love? We know it happens, but we can’t really prove that it does or how it does. Why does the universe exist? We cannot know. Because there are so many things like this that do exist but cannot truly be explained, it is very easy to want to believe in things (like Plato does) that cannot be proven.