You are right, the difference between them is very small and with large $N$ will disappear. In fact most people (at least in my experience) are not aware of any of this; "Cohen's $d$" is often used generically, many people have not heard of Hedges' $g$, but they use the latter formula and call it by the former name. The difference is that Cohen used the maximum likelihood estimator for the variance, which is biased with small $N$, whereas Hedges used Bessel's correction to estimate the variance. (For more on this topic, it may help you to read this CV thread: What is the difference between N and N-1 in calculating population variance?) The corresponding formulas are often known as the population formula for the variance and the sample formula. Recall that these are:
\begin{align}
\text{Var}(X)_\text{population} &= \frac{\sum (x_i-\bar x)^2}{N} \\
~ \\
~ \\
\text{Var}(X)_\text{sample} &= \frac{\sum (x_i-\bar x)^2}{N-1}
\end{align}
As $N$ increases indefinitely, these two estimates will converge to the same value. However, with small samples, the population formula will underestimate the variance because it does not take into account the fact that the mean, $\bar x$, was estimated from the same dataset. When these estimates are subsequently used to estimate the standardized mean difference, that implies that the former will overestimate the effect size.
Thus, with small samples, Hedges' $g$ provides a superior estimate of the standardized mean difference, but the superior performance fades as the sample size increases.
