Ground-state and finite-temperature properties of the square-lattice $S=1/2$ Heisenberg model with antiferromagnetic nearest- and next-nearest-neighbor interactions ${(J}_{1}\ensuremath{-}{J}_{2}$ model) are investigated by a spin-rotation-invariant Green's-function theory, where a reasonable agreement with numerical diagonalization data is found. The quantum phase transitions from the N\'eel and collinear phases into a spin-liquid phase are obtained at ${(J}_{2}{/J}_{1}{)}_{{c}_{1}}=0.24$ and ${(J}_{2}{/J}_{1}{)}_{{c}_{2}}=0.83,$ respectively, which considerably improves the results by a previous similar approach. The low-temperature magnetic susceptibility at ${J}_{2}{/J}_{1}\ensuremath{\lesssim}0.5,$ the temperature of the susceptibility maximum, and the antiferromagnetic correlation length are found to decrease with increasing frustration. For high-${T}_{c}$ cuprates the relationship between the $t\ensuremath{-}J$ model and an effective ${J}_{1}\ensuremath{-}{J}_{2}$ model is analyzed.