Well yes, if these things represented different things, then the output of the program would be incorrect.
To put it another way, the original version was explicitly calculating the function ℤ -> { "", "fizz" }. The new version had the results of this function directly embeded in the source code (and hence precalculated as opposed to calculated at runtime). I just don't see the two of these approaches really being all that different
I don't think the point of the post was that it was a radically new approach, but rather that it is an application of concepts that are considerably more general. The hope in using such concepts is that, being so general, you will start to see them everywhere, and using higher-level abstractions lets you see connections that you would otherwise miss. It's a kind of compression of the conceptual space that one works in. Category theory is all about trying to recognize and catalog these connections.
To put it another way, the original version was explicitly calculating the function ℤ -> { "", "fizz" }. The new version had the results of this function directly embeded in the source code (and hence precalculated as opposed to calculated at runtime). I just don't see the two of these approaches really being all that different