Power=volts * amps, so the 24 VDC motor would only require half the current
of the 12 VDC motor for a given power level. Now, the more current you
draw, the quicker the battery will be depleted and the fewer total amp hours
a battery can supply; conversely, the less current drawn, the longer the
battery will last and the more total amp hours it can supply.

Also, lower voltage and higher current means greater I^2*R losses
(simplistically, "friction" losses in the circuit due to the flow of current
and proportional to the square of the current so you get four times greater
losses for every doubling of the current), which calls for use of thicker
(more expensive) wire, and a small series of higher rated, physically
larger, and more expensive batteries. In contrast, higher voltage and lower
current allows for use of thinner (cheaper) wire, and you can use a greater
series of lower rated, physically smaller, and less expensive batteries.

The wire size issue is true for interconnecting wires (such as between
battery pack and speed controller, etc.) as well as within the motor, so a
higher voltage motor probably won't be significantly larger/heavier, and may
even be smaller/lighter, than a lower voltage motor of the same wattage.
That could even make it cheaper, all other things being equal.

Consequently, I'd suggest you avoid the 12 VDC size for sure, and consider
something higher voltage such as 36 or 48 VDC. My own electric mower uses a
60VDC motor and runs surprisingly well off a string of 5 small 12V gel
cells, easily mowing a 120'x75' lot on a single charge. At 60V, it takes
only 1/5 of the current a 12V motor would require from its 12V battery (note
that this means I^2*R losses at 60V would be 1/25 as much, if using the same
size wires, or the 12V circuit would need wires with 1/5 the resistance to
keep the same level of losses). The 12V battery would have to be sized to
provide something greater than five times the amp hours. Bigger plates,
heavier, more expensive. On the other hand, a minor savings in battery
interconnects and more readily available chargers.