As I understand it, landau roofs get their name from landaulets, where the drivers' compartment (front bench) has a fixed roof but the rear passengers have a retractable one. A Landau top imitates this by putting vinyl over the rear part of the cabin.
See:


https://en.wikipedia.org/wiki/Landaulet
Landau top:



"Vinyl roof" is a sort of umbrella term. Most of the fake landau tops we see today are vinyl, so it makes sense to call them vinyl roofs from time to time. However, something like this, which puts vinyl over the front seats in addition to the back seats (in imitation of a full convertible) shouldn't be called a landau top.


Above is my old chevy powered 262c. It has a vinyl roof, but not a landau roof. (was featured on NPOCP a while after I got rid of it)
PS- I still have the crown badges that go on the C pillars if anyone is interested.

PPS- I may very well be wrong in making this distinction, but to me it makes perfect sense looking at the etymology of "landau".