Basically, the difference is that in a functional programming language, words like "variable" and "function" have the same meaning they do in mathematics.

In C or Python, a "variable" is a storage location in which you can place different values at different times. This is completely different from mathematics, where "x" doesn't equal something different just because you progress to the next step of computation. (But "x" might refer to something different in a different context, such as when evaluating a function with a different argument; that's the sense in which it varies.)

Similarly, in mathematics, there's no such thing as a function that returns something different each time you evaluate it, or that gives one result on Wednesday but a different result on Friday, or that even gives no result at all but causes letters to appear on a nearby computer screen. Yet in C or Python, all of these things are called functions.

At first, it might seem surprising that you can do much programming without the C or Python version of functions and variables. But:

1. You can do more than you think without reassigning variables or using functions that aren't "real" functions in the sense of mathematics, and

2. Of course, in the end, all languages - including functional languages - do have a way to create boxes to store values over time, or to perform actions that make things appear on computer screens. It's just that in a functional language, these things are not the basic bread and butter tools of programming to the extent that they are in C or Python. So you have variables and functions (in the mathematical sense), and you also have other things that act like storage locations and actions, but you only use the latter when you have a real need for them.

That's the general character of the difference. There is, of course, a lot of detail that I am leaving out in a response on a web site.

The reason people were speculating about whether this would work better with people not yet accustomed to programming is that it's often thought that experience in an imperative language (like C or Python) leads you to naturally try to do things with actions and storage locations, even when it's not necessary... and then it can get frustrating and confusing to work in a language where most things are actually not done that way. There's a theory that says if you maybe learned the functional way first, you wouldn't find it as frustrating.

I'm of two minds about that. I spent a year teaching Haskell with this system (I'm the Chris Smith mentioned in the article), and I found that it's partly true that students don't get as frustrated... sometimes. But there are other cases where students naturally fall into trying to do things in imperative ways, despite not having past experience in a programming language. I actually think, computer programming aside, that this is one of the things that trip up kids in math classes. So it's not just programming languages where this comes up: a lot of teachers present math as step-by-step processes, as well. Essentially, they introduce imperative programming without the computer. This is what happens when teachers tell students that parentheses are about what to do "first", instead of about which sub-expressions are meaningful on their own. Partly, I think this is just how the human mind works, too, and it's an inherent tendency that we have to overcome to really "get" algebra and other mathematics in a deeper sense than knowing how to do specific problems. It's still frustrating, then, for kids to adapt to having to describe processes using equations and expressions, which at their core do not list things to do, but instead describe and state facts about relationships between quantities.

I still think it's an important skill to learn. Indeed, I'm a lot more interested in teaching that kind of reasoning than I am about teaching computer programming. That's why I made changes in the new CodeWorld (the site discussed in this article) that are designed to deliberately thwart students who try to think of their programs as sequences of actions, and encourage those who think in terms of compositional expressions.