One of the equations refers to the heat change (energy change) and the other refers to the absolute heat capacity of a fixed piece of apparatus.
If you already know the characteristics of the apparatus and you know how much its temperature changes for a given amount of energy then you can factor it into the equation to find out the total energy change without needing to know the mass.
For example a flask may change by ten degrees for every energy input of 1000kJ. The mass of the flask itself never changes, and it will always need this energy to change its temperature by 10ºC. It has a heat capacity (c') of 100kJ per ºC.
If you carry out a reaction using 100cm3 of solution in this flask then the total energy change will be:
E = mcdeltaT (solution) + c'deltaT flask
It is impossible to carry out the reaction without using the flask therefore this is one means of finding out the energy that the flask also absorbs.
Effectively, in any experimental energy determination the total energy will be equal to the mass of the each object that gets heated up x the specific heat capacity of each object x the temperature change of each object.
delta E = m1c1deltaT + m2c2deltaT + m3c3deltaT etc
By calibrating (knowing the energy needed by each object per degree Celcius) you can factor in the other components.
For the IB this isn't specifically (sic) required, but it can't do any harm to know. It shows that you understand the fact that energy heats everything up indisciminately in an experiment, and you may be asked to comment on errors in an experiment.
In thermodynamics experiments heat is used to change the temperature of the reactants, the container, the surroundings, the thermometer, to change liquid into vapour etc etc. Clearly any experimental values obtained using the solution alone will always be very much on the low side.