A marriage is a long term relationship between two individuals. A wedding, on the other hand, is the ceremony of getting married.
Therefore, You are invited to my wedding. is correct and You are invited to my marriage. is incorrect usage.
Their marriage was a disaster. implies that the couple was not happy in their life together and are probably separated or divorced. On the other hand, Their wedding was a disaster. implies that something happened during the wedding and that the ceremony did not go smoothly. The couple can go on to have a long and happy marriage even after a disastrous wedding.