Society of Anglo-Saxon England was both advanced and cultured.
Society of Anglo-Saxon England was not primitive or cultured.
As we know that the said society was not only advanced but also cultured, the first sentence properly expresses the statement. The second sentence expresses that the society was not primitive, which to some extent means the same as advanced, but at the same time it says it was not cultured, which isn't true. Here, it should be "not uncultured", which means the same as cultured.
He is intelligent and active. How come we can say "he is not unintelligent or active"? The correct alternative sentence should be "he is not unintelligent or inactive".
Even if we correct the second sentence by replacing "cultured" with "uncultured", I think the first sentence will still express the idea in a better way.