Recently, carbyne chain, the one-dimensional sp-hybridized carbon allotrope in the form of either α-carbyne (polyyne) with alternating single and triple bonds or β-carbyne (cumulene) with repeating double bonds, has attracted more and more attention. However, the mechanical and thermal properties of individual phase, their phase transition dynamics and defect formation remain largely unknown. Our molecular dynamics simulations show that the critical temperature for the phase transition from cumulene to polyyne is 499 K, and the phase transition is ultrafast and completed within 150 fs. To achieve perfect polyyne, however, refined temperature control is needed so as to avoid defective bonds. The bending stiffness and Young's modulus of cumulene are significantly higher than those of polyyne, while both of them are comparable to the hardest natural materials. The large difference in the stress-strain behavior between cumulene and polyyne provides a novel route for storing mechanical energy. Furthermore, the thermal conductivity of cumulene is found to be two times higher than that of polyyne, and the defective bonds can dramatically decrease the thermal conductivity of polyyne to only 13% of that of pristine polyyne. The significant changes in the mechanical and thermal properties between the two phases of carbyne chain present a great opportunity for its use as strain sensors, mechanical connectors and mechanical/thermal energy storage devices.