In this paper, we explore the thermocapillary liquid bridge between two disks of unequal radii with Prandtl numbers Pr of 0.0258 (mercury) and 0.068 (gallium arsenide) to gain insights into the underlying instability mechanism. In the context of Legendre's spectral element method, we determine the critical conditions via linear stability analysis, and then identify the instability mechanism through energy analysis. For the mercury bridge ( Pr=0.0258), our analysis suggests that the flow instability undergoes an oscillatory bifurcation for radius ratios in the range 0.5{less than or equal to} Γr{less than or equal to}0.66, whereas three transitions between two-dimensional steady axisymmetric flow and three-dimensional stationary flow can be produced by further increasing the radius ratio to 0.73{less than or equal to} Γr{less than or equal to}0.76. For the gallium arsenide liquid bridge ( Pr=0.068), the instability is always an oscillatory bifurcation in the whole computational interval. Furthermore, our observations identify six instability modes with different mechanisms. All instability modes in the mercury bridge ( Pr=0.0258) are purely hydrodynamic, but the thermocapillary mechanism cannot be ignored in the gallium arsenide liquid bridge ( Pr=0.068) because of the enhanced Pr effect.