Simplified response....
On AM, generally the whole tower is the antenna. The tower may ice over, but, unless there is so much ice it coats insulators on the guys or the base, AM works pretty much the same with rain, snow, ice or droughts. (Shunt fed towers are less effected, as they are not insulated from ground.)
On FM, only relatively small antenna elements up on the tower radiate. If they ice over so much as to change their dimensions or proportions, the signal can be affected. And if the ice creates a water path between parts of the antenna and the mount and tower, the signal can also me affected. At some point, high standing wave will shut the transmitter down.