Does something that glows red emit more heat than something equally hot that isn't glowing? Watching carbon-ceramic brakes on race cars on YouTube got me started. Heat something and its temperature will go up, the hotter is gets, the higher the frequency of the energy it emits, and so something that is hot enough to glow red is going to be emitting higher energy photons and more energy. But what about flux density? Could a different material with a lower temperature be emitting more energy? (Let's assume similar areas / volumes) Does everything start glowing at the same temperature (let's exclude things that don't survive high temperatures)?
 A: A material with a lower temperature can emit more energy than a higher temperature material, but the emission has to be non-thermal.
Any material in local thermodynamic equilibrium will emit radiation, if that material is sufficiently optically thick (ie. opaque, photons must scatter off many atoms before making it to the surface) then the material radiates like a black-body and the radiation will have a thermal spectrum, the Planck Function:
$$ 
B_\nu(\nu,T) = \frac{2 h \nu^3}{c^2} \left[ \exp\left(\frac{h\nu}{k_B T}\right)-1\right]^{-1}\ .
$$
This gives the power per unit surface area per unit frequency emitted by a blackbody. The frequency is $\nu$, the temperature is $T$, $h$ is Planck's constant, $c$ is the speed of light, and $k_B$ is Boltzmann's constant.
This spectrum peaks at a particular frequency that is linearly dependent on the temperature $\nu_{peak} \propto T$.  It is an increasing function of temperature, so a hotter (black body) material emits more energy at every frequency. The colour we associate with a hot object is simply the frequency at which the emission is peaked.  Two black-bodies with the same temperature will have identical spectra, and appear the same colour.
The total power per surface area emitted by a black body is given by the Stefan-Boltzmann Law:
$$
F = \sigma_{SB} T^4\ .
$$
This increases very strongly with temperature.  A hot object with a thermal spectrum emits a lot more energy than a cool object with a thermal spectrum, every time.
However, not everything emits thermally!  LEDs, for instance, have a very non-thermal spectrum. So do fluorescent lights and lasers.  
So, your answer: if the materials are emitting thermally, then the hotter object will always emit more energy per surface area.  If they are not emitting thermally (an IR laser for instance), it's anyone's game!
