On their respective power supplies both motors should perform identically but the 12V motor would draw twice as much current from its 12V supply compared to the 24V motor on a 24V supply.
In other words power supplied to both motors should be quite similar for a given mechanical load. Mechanical load power is defined as \$2\pi n T\$ where n is speed in revs per second and T is torque is newton-metres. And if the 12V motor took 4 Amps to supply a certain mechanical power output then the 24V motor would take 2A to perform identically.
DC motors of the simple type (trying not to generalize here) will rotate at full speed on no-load and this speed is mainly determined by the applied voltage. Putting 24V on a 12V motor may wreck it. Conversely, on heavy loads, the speed normally reduces fairly linearly with torque but on a 24V supply there may be enough potential for the motor to fry due to it being able to supply more torque and speed. Don't do it is my advice.
I don't understand what is happening on this diagram.
Referring to the diagram in OP.
This MOSFET-TRANSISTOR duo is used to enable and disable the power supply of the module, when MCU_CTRL is high Q101 is enable, which in turn enables the Q102 by pulling the GATE of the PMOS Q102 to GND, and enable the power for module on VBAT pin. When MCU_CTRL is low, in similar fashion it disable the power in VBAT pin.
For one thing, the part I was most worried about, the capacitor, seems not to be listed. The description of the diagram states that "A low ESR tantalum capacitor is usually used. The value for the capacitor should be more than 470uF."
The datasheet of SIM900 suggest low ESR 100uF capacitor(tantalum will be good) with a small 0.1uF or 1uF ceramic in parallel placed close to the SIM900, on page 20 of datasheet.
It also says that "When there was no on/off pin in the LDO or DC-DC IC, the customer can follow nether reference schematic to control the VBAT on/off." which seems to imply this doesn't actually show all of what goes into the VBAT and there's more to the circuit outside the VBAT_IN.
This particular section in the datasheet gives you an option to enable/disable the power supply of SIM900 using the MOSFET-TRANSISTOR duo if in case there is no ON/OFF pin present on the DC-DC converter or LDO. So either optios you can use to enable/disable the power supply of SIM900. But the datasheet of SIM900 shows it has got PWRKEY which is default pulled-up to enable or disable the power supply. Ref: Datasheet page 17.
I'm also confused as to what the MCU_CTRL is
The MCU_CTRL is external pin coming out from the master controller/processor, not coming from SIM900 to enable/disable the power supply.
In the starting of the OP, it disusses about powering the SIM900 using RBBB arduino which can supply a maximum of 250mA to 300mA which is using L4931 LDO.
But I am afraid it is not the correct approach to power SIM900, since the datasheet clearly expects to power SIM900 using LDO or DC-DC Converter which is capable of 2A current, with a minimum voltage of 3.4V. Ref: Datasheet page 20
Best Answer
Looks to me like it has a switching power supply and can operate from either 5V or 12V. It's likely it might work okay on other (intermediate) voltages but it's not specified for operation other than at 5.0V+/-10% or 12.0V+/-10%. From the data:
Also:
That said, I would have a gander at the PCB near the input power connector and switching power supply circuitry to see if there is some kind of jumper or something like that before applying 12V to the module. That's in the upper left of the PCB photo and I see nothing but the photo is quite blurry.