A terahertz metamaterial structure consisting of two U-shaped split-ring resonators and a horizontal cut-line resonator is designed for realizing the electromagnetically induced transparency (EIT) effect and the Fano resonance effect. The genesis of the EIT and Fano resonance bifunctionality is illustrated by combining the near-field distribution analysis of transparent windows and transmission dips. Interestingly, the bifunctional dual-band transparency effect could transform into a single-functional single-band transparency effect when we change the position of the square metal sheet in the vertical direction of the composited terahertz metamaterial structure. In addition, based on the high-quality factor of Fano resonance response, the proposed bifunctional terahertz metamaterial device has good refractive index sensing sensitivity. These results indicate that our proposed terahertz metamaterial can provide guidance for the design of subsequent multifunctional and integrated metamaterials and optoelectronic devices.