If you have two independent random variables that are normally distributed (not necessarily jointly so), then their sum is also normally distributed, which e.g. means that its excess kurtosis is $0$.
On the other hand in case the mixture of one-dimensional normal distributions the mixture distribution can display non-trivial higher-order moments such as skewness and kurtosis (fat tails) and multi-modality, even in the absence of such features within the components themselves (see also this video for an easy enough example). This means that the result doesn't have to be normally distributed.
How can you reconcile these two results and what is their connection?