As you get to smaller scales, microscopic changes in flow conditions will 'trip' a flow into turbulence. With low energy flow, small changes in flow conditions will not cause a destabilising effect. As flow energies increase, smaller and smaller discrepancies cause molecular diffusion to become a significant factor. This is why turbulence is not connected directly to Reynolds number. The flow is dependent on the exact situation and conditions.
The reason why it was so difficult to understand for years is that the changes simply could not be seen or measured. So you had one flow that looked like it had exactly the same conditions as a second flow, one was turbulent and the other was laminar. This was because on a scale smaller and quicker than was measurable the flow conditions were altered. With computer simulation direct solution you can slow 'time' down to see the point where turbulence is tripped.
We certainly don't understand turbulence fully, but it wasn't understood nearly as well in Feynmans time because they simply didn't have the technology to see or try to predict what was going on. To an extent we still don't, it's only just getting tothe point where we can directly solve the equations that govern a flow and they replicate reality to the highest degree of accuracy possible at this time.
As an analogy it's similar to gravity and weak force. On large scales gravity dominates. in fluids viscous and inertia effects dominate. When you get to very very small scales, weak force dominates. In fluid flows molecular attraction and diffustion effects donimate over viscous and inertia effects.
(although they are acutally understood) :D
EDIT: Although we suspect that diffusion effects casue turbulence, we don't know the exact mechanism or how or why. (Which I suppose is more to the point of your original question).