We study approximate maximum likelihood estimators (MLEs) for the parameters of the widely used Heston stock and volatility stochastic differential equations (SDEs). We compute explicit closed form estimators maximizing the discretized log-likelihood of $N$ observations recorded at times $T,2T, \ldots, NT$. We study the asymptotic bias of these parameter estimators first for $T$ fixed and $N \to \infty$, as well as when the global observation time $S= NT \to \infty$ and $T = S/N \to 0$. We identify two explicit key functions of the parameters which control the type of asymptotic distribution of these estimators, and we analyze the dichotomy between asymptotic normality and attraction by stable like distributions with heavy tails. \\ We present two examples of model fitting for Heston SDEs, one for daily data and one for intraday data, with moderate values of $N$.