This paper is devoted to optimal functional inequalities for fractional Laplace operators on the sphere. Based on spectral properties, subcritical inequalities are established. Their consequences for fractional heat flows are considered. These subcritical inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities. Their optimal constants are determined by a spectral gap. In the subcritical range, the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. We also consider inequalities which interpolate between fractional logarithmic Sobolev and fractional Poincaré inequalities. Finally, weighted inequalities of Caffarelli-Kohn-Nirenberg type involving the fractional Laplacian are obtained in the Euclidean space, using a stereographic projection and scaling properties.