Why is free radical chlorination exothermic whereas bromination is endothermic? In the energy profile diagrams below, it can be clearly seen that the free radical chlorination of an alkane

$$\ce{RH + Cl2 -&gt; RCl + HCl}$$

is exothermic. However, the corresponding bromination

$$\ce{RH + Br2 -&gt; RBr + HBr}$$

is endothermic. Why is there a difference?
First off, we can see that both reactions are exothermic If we do the thermodynamic analysis.

Given that your starting materials have the same heat of formation: 

$$\Delta H^\circ_f(Products) = \Delta H^\circ_f (\ce{R-H}) + 0, \quad{\Delta H^\circ_f(\ce{Cl}) = \Delta H^\circ_f(\ce{Br})= 0}$$

$$\ce{R = Et}\\ \Delta H^\circ_f (\ce{EtH}) = \pu{-84 kJ mol^{-1}}$$
$$ 
\begin{array}{cccccc}
\hline
\text{Product 1} &amp; \Delta \text{H}^\circ_f [\pu{kJ mol^{-1}}] &amp; \text{Product 2} &amp; \Delta \text{H}^\circ_f [\pu{kJ mol^{-1}}] &amp; \Sigma\ \Delta \text{H}^\circ_f\  [\pu{kj mol^{-1}}] &amp;\text{Result}\\
\hline
\ce{HBr} &amp; -\text{36.45} &amp; \ce{Et-Br} &amp; -\text{97.6} &amp;-\text{134.05}&lt;-\text{84} &amp;\text{Exothermic} \\
\ce{HCl} &amp; -\text{92.31} &amp; \ce{Et-Cl} &amp; -\text{137} &amp; -\text{229.31}&lt;-\text{84}  &amp;\text{Exothermic}\\
\hline
\end{array}
$$



Back to your question:


  Why is there a difference?


The graphs you are looking at only consider the reaction:

$$\ce{R-H + X^. -&gt;  R^. + HX}$$

For which we get:
$$ 
\begin{array}{cccccc}
\hline
\text{X}
&amp; \Delta \text{H}^\circ_f(\ce{X^.})
&amp; \Delta \text{H}^\circ_f(\ce{RH})
&amp; \Delta \text{H}^\circ_f(\ce{R^.})
&amp; \Delta \text{H}^\circ_f(\ce{HX})
&amp; \Delta \text{H}^\circ_{rxn}
&amp; \text{Result}\\
\hline
\ce{Br} &amp; -\text{96.94}^{**} &amp; -\text{84} &amp; -\text{119}^\dagger &amp;-\text{36.45} &amp;\text{25.49}&amp;\text{Endothermic} \\
\ce{Cl} &amp; -\text{121.29}^{**} &amp; -\text{84} &amp; -\text{119}^\dagger &amp; -\text{92.31} &amp;-\text{6.02}&amp;\text{Exothermic}\\
\hline
\end{array}
$$
*all units in $[\pu{kj mol^{-1}}]$ all values from Wikipedia unless otherwise stated 
** Table of Bond Dissociation Energies
$^\dagger$ Values from NIST WebBook

Here the enthalpy of radicalization of the alkane is different for chlorine or bromine, but the halogenation in total is exothermic