I recently saw them in a grad eng class, but I agree with the article - from what I saw there is no need to give them the math professor treatment. You can use them as a piece of trivia - ie pde of type x has this set of basis functions - now apply the principles of basis functions to solve your your problem.

But are they useful now, other than as nomenclature? Bessel functions are defined as the solutions of Bessel's differential equation. It's all a bit circular. (there's the series expansion, but it doesn't gain you much)

30 years ago, if I wanted to plot the result of solving an equation like this, Bessel functions were useful as I'd just reach for Abramowitz&Stegun and look at the tabulated values. But now I have a computer, tabulated special functions don't matter nearly so much.

It's a long time since I had to use Bessel functions, so I could be very wrong, but this might be one of the reasons Rota said that.