The design and safety of nuclear reactor are strongly dependent on the neutronics and thermal conduction. The interaction between these coupled issues forms multi-physics nonlinear system, which has limited the economy of the nuclear power plant because of the complexity. And the accurate prediction of the reactor behavior still remains a challenge for the nuclear engineering. Traditionally, the neutronics and thermal conduction are solved separately by operator-splitting, which may be unstable and introduce significant errors. In recent years, fully coupled Newton-Krylov methods are prefer to be adopted due to these methods are more robust, accurate and efficient. However, each Newton-Krylov method has merits and demerits. In this study, finite difference Jacobian based Newton-Krylov (DJNK) and widely used Jacobian-free Newton-Krylov (JFNK) method are employed to solve the coupled neutronics/thermal conduction problems of a high-temperature gas-cooled reactor (HTGR). A steady state at thermal power of 250 MW and a transient state of supercritical accident are simulated. The results show that, DJNK is more efficient compared with JFNK, because DJNK performs better in preconditioning, which is the key to successful application of Newton-Krylov methods. And different preconditioning methods are utilized to reduce the number of Krylov iterations. A speedup ratio of DJNK over JFNK is observed to be 3. Synthesis results indicate that compared with JFNK, DJNK is a more potential method, contributing to a deeper understanding of the reactor behavior in a more efficient way.