What is the heat needed for an isothermal expansion of gas. Why do physics and chemistry yield different answers?
The problem is that both the problem statement you presented and the chemistry solution you showed are fallacious.  The fallacy is that, for an ideal gas $V=nRT/P$, so that, if the pressure and temperature are both constant, the volume can't change.  If the volume does change, then either the pressure must change or the temperature must change.  

In the "chemistry solution" you presented, the pressure is taken as constant (atmospheric).  So the temperature must change.  But if the temperature changes, then $\Delta U$ is not zero (as they assume).  So the solution is incorrect.  The correct solution, based on the 1st law, is $$\Delta U=nC_v\Delta T=Q-P_{atm}(V_2-V_1)$$or equivalently, $$\Delta H=\Delta U+P_{atm}(V_2-V_1)=nC_p(T_2-T_1)=Q$$Also, since $$n=\frac{P_{atm}V_1}{RT_1}$$we have $$\frac{T_2}{T_1}=1+\frac{QR}{C_pP_{atm}V_1}$$

In the "physics solution" you presented, they make the opposite assumption that, rather than the pressure remaining constant, the temperature remains constant.  For the temperature remaining constant, the solution you presented is correct.  But the final pressure is:  $$P_2=P_{atm}\frac{V_1}{V_2}$$