What is the difference between Gaussian White noise and $iid$ noise and how can I check?

Question

If I understand correctly, a series {$X_t$} is $iid$ noise if there is no trend or seasonal component and the observations {$x_t$} are independent and identically distributed with zero mean, while a series {$X_t$} is Gaussian White noise if it is a sequence of uncorrelated, i.e. $\rho(X_i,X_j)=0$, where each $X_i$ has zero mean and variance $\sigma^2$. But how can it be that every $IID(0,\sigma^2)$ sequence is also a $WN(0,\sigma^2)$ sequence, but the converse is not true (independence vs uncorrelated)? And can someone please name some tests that should I use when checking for if a 1000x1 vector time series is just White noise or also $iid$ noise? Thanks!