In this paper, the spin wave theory is applied to the Heisenberg spin bilayer with intralayer ferromagnetic interaction J1, intralayer single-ion anisotropy D and interlayer antiferromagnetic interaction J2. It is found that the effects of both D and J2 on the thermodynamic quantities give rise to the two different low-temperature asymptotic behaviors with and without exponential law. For D>0, the interlayer antiferromagnetic interaction can induce the appearance of the maximum of the layer magnetization at finite temperatures. At the location of the layer magnetization maximum, the approximate behaviors (such as the power, linear, rational, exponential and logarithmic laws) which are driven by the temperature or the anisotropy, are obtained for the low-temperature thermodynamic properties. It is shown that the presence of antiferromagnetic interlayer interaction J2 clearly induces more quantum fluctuations than the case where the interlayer interaction J2 is ferromagnetic. Our results of the layer magnetization agree with the experimental data of the layered van der Waals crystal FeCl2 at low temperatures. In the monolayer case of J2=0, our results are in agreement with the findings obtained by the existing theories and the quantum Monte Carlo data.