I think that was a fairly idiosyncratic point of view of Feynman's. In actual scientific practice you can find hundreds of examples of published scientific papers that address 'why' questions. Here are a couple of completely random examples:
They answer the why's with the same way @hansbo complained about not answering the why, e.g:
" We show that the symmetries of this non-commutative space unify the standard model of particle physics with (2)
chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time."
It's not uncommon for a scientific paper to raise a question without fully answering it (science is hard). The point is that actual scientific practice does not appear to care about any distinction that can usefully be described as a distinction between 'how' and 'why' questions. You can keep asking 'but why?' ad infinitum and never arrive at a fully satisfying explanation. However, the same is also try of 'but how?' We will find no ultimate answers, but the questions that stimulate scientific research certainly seem to include 'why' questions.
" We show that the symmetries of this non-commutative space unify the standard model of particle physics with (2) chiral gravity. The algebra of the octonionic space yields spinor states which can be identified with three generations of quarks and leptons. The geometry of the space implies quantisation of electric charge, and leads to a theoretical derivation of the mysterious mass ratios of quarks and the charged leptons. Quantum gravity is quantisation not only of the gravitational field, but also of the point structure of space-time."