Eating fish and meat together is not restricted by Biblical law. However, the Rabbis considered that mixing fish and meat together could be harmful for one's health (סכנה). The Gemara Pesahim (76b) is not explicit about the particular health hazard meat and fish together would trigger. It describes the problem as משום דקשיא לריחא ולדבר אחר . dabar aher, which literally means "something else" is a term which is usually used in the Talmud as a euphemism. It might be indicating thus that the rabbis were describing some type of disease which we are not able to identify today.
Although Maimonides does not mention this prohibition, the shulhan 'aruh does (Yore Dea'a 116:2). Rabbi Yosef Karo writes about it in the chapter where he deals with hazardous foods. Following Rashi's opinion, Rabbi Yosef Karo identified the harm triggered by this mixture as tsara'at. Now, we are not certain what type of disease tsara'at was. And although tsara'at is traditionally understood as a type of leprosy Maimonides explained that tsara'at should not be identified narrowly as leprosy but in a more broadly sense, probably as a general skin disease (MT Tsara'at 16:10). It is possible that eating fish and meat together might cause in some people an allergic reaction. Perhaps, as the famous commentator of the shulhan 'arukh Magen Abraham said, in our days human nature has changed (נשתנו הטבעים) and people are not affected anymore by this mixture as they used to be affected centuries ago.
Other rabbis have a completely different interpretation about the physical risks of eating meat and fish together. In their opinion it has to do with the possibility of swallowing a fish-bone hidden in a piece of meat.
In any case, and although contemporary rabbis have not reached a definitive conclusion as to what is (or what was) the physical danger caused by eating fish and meat together, we strictly uphold this restriction, following the rulings of the shulhan 'arukh (קבלנו הוראות מר"ן). It is important to clarify that unlike milk and meat, fish and meat may be eaten one after the other at the same meal as separate courses.