This article studies the weak convergence of discrete-time affine stochastic volatility models driven by both Gaussian and non-Gaussian innovations. Our results generalize the existing diffusion limits for affine GARCH models and provide new insights on their relationship with continuous-time stochastic volatility models. Notably, we show that the canonical affine volatility models popularized in discrete and continuous time are not the analog of one another from the point of view of weak convergence.