Muslims are very much about follow the rules and you go to Paradise when you die. The religion is designed to be rule followers. Mohammed was the political, military, and religious leader of his time so there was no separation and that non-separation has continued.
In Christianity, it is about choice. Granted some sects have veered from that over the centuries but for the most part it is a personal choice to believe. In fact, New Testament actually says you cannot believe for someone else. Therefore western culture evolved to independent choice with a respect to the 10 commandment laws.
Democracy gets along much better with independent christian-like thought processes. Muslim thought processes are about obeying the rules set down by others and being deferential to the leader who is often ruling at the discretion of the religious leadership. It is hard for people that have been raised for centuries to do what they are told to now act independently. This will happen over time but the birthing process is painful.
I don’t think that Islam and democracy are 100% at odds but I do believe that the separation of state must be allowed. The religious community cannot dictate to the political community. The reverse of this is true. One person cannot believe for another therefore laws and processes that dictate belief are ultimately doomed to failure.