The main advantage of this approach is that you can use any C++ compiler and the library (in this case Eigen) will work. The disadvantage of this approach is that the code is incomprehensible to average scientists, you must spend a lot of effort to become a C++ expert.

The advantage of Fortran is that the code is simple and comprehensible to average scientist, and yet fast, because the effort goes into the compiler itself. I would also argue it is easier to implement an optimization pass in the compiler than implement optimizations on the template level in C++. The disadvantage is that you need a good Fortran compiler. If your compiler can't run your code on, say, GPU, then suddenly there is not a good path forward. While in C++, there is always a way forward via template metaprogramming: perhaps ugly, but at least it gets the job done.