Difference in cut-off point when powering camera from 24v output vs 14.4v output
Consider a typical NiMH single cell battery discharge curve (http://data.energizer.com/pdfs/nickelmetalhydride_appman.pdf)



Lets assume your 14V battery is 12 cells and your 28V battery is 24 cells in series.  With a nominal 1.2V/cell for a NiMH battery, that gives 14.4V and 28.8V as the nominal battery voltages.

In the 14V case, your cutoff is 12.5V, or 1.04V per cell.  Referring to the graph, that occurs at maybe 95-98 % discharged.

In the 28V case, your cutoff is 21.5V, or 0.9V per cell.  According to the graph, that's probably 98-99% discharged.

What if you discharged the 28V battery all the way to 12.5V, or 0.52V per cell?  Well, first, notice that the voltage drops off almost instantly between 0.9V and 0.5V - you won't get any significant extra battery life.  But also, keep in mind that each cell will discharge slightly differently.  If one cell has discharged to 0V while the other cells are still around 0.7V, and you keep discharging all the way down to an average of 0.5V per cell, you will start to reverse charge the weak cell, which can damage it.

TL;DR - discharge limits are typically set on a per cell basis to prevent the batteries from over-discharging, which can damage them.