For jointly (per @Did) normal random variables, uncorrelated implies independent.
In particular, it is easy to see that the joint density function factors,
giving the product of the two marginal density functions.
Also, for normal data, the sample mean $\bar X$ and sample SD $S$ are
independent. (Proof via linear algebra or moment generating functions.) But $\bar X$ and $S$ are not independent except for normal
In the left panel below $S$ is plotted against $\bar X$ for 30,000
randomly generated standard normal datasets of size $n = 5.$ As a 'naturally occurring' instance where zero correlation and dependence coexist: in the right
panel the same is done for 30,000 samples of size $n = 5$ from $Beta(.5, .5).$
For these beta data $\bar X$ and $S$ are uncorrelated, but not independent.
m = 30000; n = 5
x = rnorm(m*n); NRM = matrix(x, nrow=m)
ax = rowMeans(NRM); sx = apply(NRM, 1, sd)
## -0.001177232 # consistent with uncorrelated
y = rbeta(m*n, .5, .5); BTA = matrix(y, nrow=m)
ay = rowMeans(BTA); sy = apply(BTA, 1, sd)
## -0.001677063 # consistent with uncorrelated