One major motivator for choosing a low-order modulation scheme such as BPSK or QPSK over 16-QAM or 256-QAM is improved energy efficiency. Different modulation schemes require different Eb/N0 for a given BER. Eb is the RF signal energy per bit, and N0 is the noise power spectral density. Assuming that N0 does not depend on the modulation scheme, this simply means that the schemes require different amounts of received signal energy per bit to achieve a given BER. A lower Eb/N0 requirement thus implies that less signal energy needs to be transmitted per bit.
For better TX energy efficiency, it is thus good to use a modulation scheme that requires as low Eb/N0 as possible for the targeted error probability. From this perspective, it is good to choose BPSK, QPSK or, even better, an orthogonal modulation scheme such as M-PPM or M-FSK with a very high M. The improved energy efficiency, however, often comes at the cost of spectral efficiency.
For a battery-powered wireless device, choosing BPSK over 256-QAM would generally improve the device lifetime. On the contrary, in an application where power and energy consumption are no concern at all, choosing 256-QAM or even 1024-QAM would of course make more sense because of the higher data rate.
In contrast to what the OP says, use of higher-order QAM is generally not more energy-efficient than use of lower-order QAM -- the higher the order, the greater Eb/N0 is required.
One can reach these conclusions by reading communications literature and by performing waveform-level RF signal reception simulations using Matlab or Octave. Particularly performing the simulations is recommended for understanding the differences between modulation schemes.
Additionally note that, compared to the simpler modulation schemes, the generation of QAM signals may require more complex electronic circuitry (i.e. a greater amount of high-power-consuming circuit blocks). It is easier to achieve high TX/RX power-efficiency using OOK/BPSK/BFSK compared to 1024-QAM.