My question is, where is the new way of computing? I really do not want to criticize this language. I just want to have something that I really get an extreme advantage to gain through this language.
You open a file, do a fourier transformation and really ask, where the new way of computing is? Really? The file could be opened with fortran 66 or even with Zuse’s machine and a fourier transformation is a fourier transformation is a fourier transformation.
For me Mathematica is an algorithmic laboratory - you can use things together which are elsewhere far away from each other - look at some of the Vitaliy Kaurov posts in this forum or for example see the post Dancing with friends and enemies by Simon Woods in this forum. Try to redo that.
In the end - because of Wolfram’s own Principle of Computational Equivalence - there is no extreme advantage of Mathematica over other systems on philosophical grounds: but it matters a lot if you like or dislike a language you are using, whether you have an integrated system or not, whether you can use a problem oriented programming style or not, whether you can work on the problem inside the notebook structure or not, whether you have data at your disposal or not …
An advantage which is not so small: Reading the help by chance you stumble about stuff you probably didn’t know before.