See my answer to a related question https://stats.stackexchange.com/a/253235/71228
Yes, both methods use essentially the same assumptions and are capable of learning the exact same kind of underlying data distribution (mixture of Gaussians); the difference is the use case.
Your question is somewhat ill-posed, since you want to compare a supervised learning method (classification given class labels) to an unsupervised learning method (clustering with no class labels).
If you're talking about classification performance given class labels, and assuming somehow you could identify the class labels with clusters learned from EM (and more importantly you somehow know how many true clusters there are!), I would guess QDA would give slightly better results, since no label information is used in clustering the data with GMM+EM.