According to the in-game tooltip of Elemental Rage, the Fiery Rage version of this skill is supposed to deal 10% more damage.
Comparing the tooltips of Elemental Rage while wielding a Frost Staff (Icy Rage) and a Flame Staff (Fiery Rage) shows an equivalent damage value.
The weapons used were level cp150 crafted, setless, normal quality. During the comparison my character was gearless (except for the staff) and without food, no CP allocated, 64 attribute points into Magicka, Lover Mundus, empty skill bars except for Elemental Rage - the character's class is Sorcerer and race is Altmer, hence the elemental damage bonuses from racial passives are equal for all elemental damage types and the class damage bonuses only affect lightning damage, which is not considered for the purpose of this comparison.
Below are the tooltips as read while wielding a Frost Staff first, and a Flame Staff second.
In order to corroborate the finding, I proceeded to test Elemental Rage on a dummy with both staves, which confirmed the bug
I invite the readers to test this themselves making sure there are no passives interfering and confirm whether the above is true.