There are neither individual drivers nor is it a persistence of vision.
The displays are typically multiplexed 1/16 with 5x8 dots per character. There is not enough RAM internally to control each dot individually (this is a mid-1980s vintage chip)- the internal character generator ROM generates characters (and there are some custom characters allowed for in RAM).
Here is what the hardware connection looks like for an 8x2 display connected to an HD44780 compatible controller:
As you can see, only 56 connections allow 640 pixels to be controlled. Complex waveforms switching between 5 voltage levels allow the controller to individually control the state of each pixel above or below a threshold voltage and simultaneously maintain essentially zero DC component between commons and 'segments' (more than 50mV can damage the LCD 'glass' (the bare display, not literally the glass) by causing electrochemical action).
The disadvantage of the 1/16 multiplex scheme is that the display is temperature sensitive and tends to have lower contrast and viewing angle because it has to narrowly distinguish between slightly different AC voltages across the pixels. Hence the contrast pot or temperature compensation circuit to shift that threshold around so that the dots that should be black are black and vice versa.
There is an LGPL opencores HD44780 compatible HDL (Verilog) if you look for it.
LEDs are diodes and don't care about average voltage so driving them is simpler, but not necessarily easier. LCDs have the advantage they take almost no current, so a bunch of analog switches made of on-chip MOSFETs can switch the outputs. For a 8x2 LED matrix you might use a 40 bit power shift register to sink the column lines and source row lines with 16 high side drivers. If you want 1mA average per LED you would have to allow for a 640mA supply current, and the column lines would have to sink up to 16mA at 100% duty cycle and the row drivers source 640mA each at 1/16 duty cycle. Probably the 16mA is not acceptable using the FPGA directly because, while the individual sink current is reasonable, the total current through the ground pins is likely too high.
Edit: You asked about 16x2 display- those use an additional driver HD44100-compatible chip to handle the other half of the display. The HD44780 can only handle an 8x2 display by itself (also corrected some numbers above).