We study the effects of confinement and hydrophobicity of a spherical cavity on the structural and thermal stability of proteins in the framework of a hydrophobic-polar (HP) lattice model. We observe that a neutral confinement stabilizes the folded state of the protein by eliminating many of the open-chain conformations of the unfolded state. Hydrophobic confinement always destabilizes the protein because of protein-surface interactions. However, for moderate surface hydrophobicities, the protein remains stabilized relative to its state in free solution because of the dominance of entropic effects. These results are consistent with our experimental findings of (a) enhanced activity of alcohol dehydrogenase (ADH) when immobilized inside the essentially cylindrical pores of hydrophilic mesoporous silica (SBA-15) and (b) unaffected activity when immobilized inside weakly hydrophobic pores of methacrylate resin compared to its activity in free solution. In the same vein, our predictions are also consistent with the behavior of lysozyme and myoglobin in hydrophilic and hydrophobic SBA-15, which show qualitatively the same trends. Apparently, our results have validity across these very different enzymes, and we therefore suggest that confinement can be used to selectively improve enzyme performance.