Have you ever heard the saying "religious propaganda"? The term is used when referring to acts that are in place because of religious views - views that have very little to do with faith in God and more to do with faith in man-made churches. So basically, in my opinion, the reason why people use the term "Religiously Speaking" is because the Church is often seen as a joke in non-religious circles - whereas the term "Spiritually Speaking" refers to something that you just know through faith. So the term "religiously speaking" is used when people are wanting to find answers to silly questions that are centered from the teachings of a Church - kind of like saying "the joke is on you". You will see a lot of questions on here that start "Spiritually Speaking", these questions usually lead to questions wanting an explanation about a lesson being taught by the Church.
Hope this helps.