The seerah (biography) of the Prophet Muhammad (s) gives guidance on this distinction (between religion, here Islam, and culture) in the light of the Hijrah:
"In effect, exile (hijrah) was also to require that the first Muslims learn to remain faithful to the meaning of Islam's teachings in spite of the change of place, culture, and memory. Medina meant new customs, new types of social relationships, a wholly different role for women (who were socially far more present than in Mecca),.... Very early on, the community of faith, following the Prophet's example, had to distinguish between what belonged to Islamic principles and what was more particularly related to Meccan culture. They were to remain faithful to the first while learning to adopt a flexible and critical approach to their original culture. They even had to try to reform some of their attitudes, which were more cultural than Islamic. Umar ibn al-Khattab (ra) was to learn this to his cost when, after he had reacted most sharply to his wife answering him back (which was unthinkable in Mecca), she retorted that he must bear with it and accept it just as the Prophet did. This was a difficult experience for him, as it was for others, who might have been tempted to think that their habits and customs were in themselves Islamic: hijrah, exile, was to reveal that this was not the case and that one must question every single cultural practice, both to be faithful to Islamic principles and to open up to other cultures and gain from their wealth. For instance, having learned that a wedding was to take place among the Medina Muslims (the Ansar), the Prophet had two singing maids sent to them, for, he said, the Ansar enjoyed singing (Hadith by Ibn Majah). Not only did he recognize a cultural feature or taste that was not in itself opposed to Islamic principles, but he integrated it as an enrichment of his own human experience. Hijrah was also, then, a trial of intelligence, spurring the need to distinguish between principles and their cultural manifestations..."
extract from 'The Messenger - The Meanings of Life of Muhammad' by Tariq Ramadan (Penguin books and Oxford University Press)
