Isothermal natural gas flow model with constant gas temperature has been widely used in conventional coordinated operations of integrated electricity and natural gas systems (IEGS). However, as a crucial state variable in IEGS, gas temperature varies significantly among nodes in a natural gas system, which can be calculated along with nodal pressures and mass flow rates. This paper proposes a novel non-isothermal optimal power and gas flow (OPGF) model that avoids the isothermal assumptions involved in traditional natural gas network model and improves the accuracy significantly. In the proposed model, partial differential terms with minimal weights are neglected based on the quantitative analysis of the energy conservation equation to further simplify the energy conservation equation. A non-isothermal model of natural gas networks is developed by introducing the finite difference method and laws of thermodynamics to incorporate the gas temperature change into the coordinated operation of IEGS. A start point initialization scheme is developed based on the Newton-Raphson algorithm to accelerate the interior point method solution convergence and improve its solution quality. Numerical experiments presented for two test systems validate the effectiveness of the proposed OPGF model and solution methodology.