Well there weren't any direct rules barring using similar weapons but both sides had a different background related to warfare. The muslims come from a history of horse archery and flexible armor. The general hotter climate limited the use of heavy warm armor like plate with a padded coat underneat. You would simply die from heatstroke more often then not. The Crusaders for example started to wear tabards. The tabards covered the bare metal plates that would otherwise become scorching hot. Sure they don't directly help against the heat you feel from the sun but it does prevent you from burning yourself against your plate armor.

Armor is almost always leading to weaponry. Because of the lighter armor worn by the muslims they developed 'lighter' weaponry. Mounted archers exclude the use of large warbows like the longbow, while compound bows were dominant you still get a lower punch. This wasn't an issue against their own lighter armor but against full plate armor they were useless. There accounts of knights walking around like pincussions. Simply ignoring all the arrows sticking into their armor and bouncing off.

The same can be somewhat said for scimitars. While curved and single edged swords have always been used in Europe scimitars are better suited for cutting then stabbing against weakpoints. Now of course was a sword a sidearm but still, again the lighter armor meant the muslim weaponry was less useful against a fully armored knight.

However as mentioned before, full plate is not as heavy as often said it's still freaking hot in the sun. Under the plate you got most likely your mail or at least padding. THen under that you got clothing. That's at least 4 layers, one is insulating and another is metal.