I designed in a 1x16 character VFD display into a product back in the early 1990s. Like this one it also came from Futaba. This was a 'bare' display with no driver chip as yours seems to be.

Driving the display helps if you consider that it resembles a multi-anode thermionic valve (vacuum tube), actually a multi anode, multi grid directly heated cathode triode, where anode current results in illumination of 1 segment of 1 character. Control of the segment displayed is by 16 grids.

There are 16 anodes brought out of the display, one for each character and 16 grids, one for each segment. A common filament is the negative side of the 'tube'. In your case with 2 lines of display, there will be some method of addressing line 1 or 2 I assume.

The software to drive it is simply a typical multiplexed character and segment arrangement.

Because the display uses 'odd' (i.e. not 5V logic) voltages to drive it, I used 2x8 bit latches driving the grids and 2x8 bit latches driving the anodes. Logic level to display level voltage translation was done with 32 'digital transistors' (integrated series base resistor).

I can tell you that the task is 'non-trivial' but rather rewarding. With luck I might even be able to find the microcontroller code used (written in PL/M 51 and very readable).