Abstract: Solar differential rotation plays a key role in angular momentum transport and large-scale magnetic field evolution, yet its hemispheric asymmetry and mid-term periodicities remain poorly understood. This study uses the solar differential-rotation coefficient time series A and B, where A denotes the equatorial rotation rate and B represents the latitudinal gradient of differential rotation. Derived from Carrington synoptic magnetograms within the ±30° latitude band from 1975 to 2023, these series are used to investigate the hemispheric asymmetry and periodic evolution of solar differential rotation, as well as its phase relationships with sunspot area, through Fourier and wavelet analyses. The main results are as follows: (1) Coefficient A is positively correlated with sunspot area, whereas coefficient B is negatively correlated. Both coefficients reach their extrema several months before solar maximum, indicating a temporal lead of rotational variations over surface magnetic activity. (2) Coefficient B in the Northern Hemisphere leads that in the Southern Hemisphere by about 1 year, while sunspot area shows a hemispheric lead of about half a year. Meanwhile, the amplitudes of coefficients A and B have both declined since Solar Cycle 21, indicating a long-term weakening of solar activity and latitudinal shear. (3) Both coefficients A and B exhibit multi-timescale periodicities, including subannual, 2-3 year, and 11 year variations. In particular, coefficient B shows a pronounced QBO (Quasi-biennial Oscillation), which is stronger in the Southern Hemisphere and persists across multiple solar cycles, suggesting a close coupling between solar rotation and magnetic fields. The hemispheric QBO and north-south asymmetry identified in this study provide new observational constraints for solar dynamo models.