the first equation calculates how much a SURFACE radiates heat at different temperatures. while the second one calculates the heat change in the whole body. so its a different formula.
there is no stock here to talk about. the original meaning of the equation is that a body at temperature T will emit a heat Q. to calculate the heat loss you went to another formula here.
if the body is a black body then the equation should stand. just to check the a u mentioned in the beginning is the Boltzmann's constant?
why shouldn't there be a physical truth behind the equation? a small change in temperature will only change the amount of energy emmitted by a fraction...