This comparison evaluates the performance of Elemental Shamans and Frost Mages, highlighting their strengths and weaknesses. Using SimulationCraft, each spec was tested under various conditions to analyze their capabilities in level 80 end-game PvE gameplay. Updated for patch 11.0.5 of the The War Within expansion.
Performance ComparisonPatch 11.0.5
Frost Mage+5.21% DPS
When simulating for maximum theoretical single-target DPS, Frost Mages averaged 5.21% more DPS than Elemental Shamans (+90K DPS).
Elemental Shaman+26.92% DPS
When simulating for light AoE DPS potential, with small waves of adds, Elemental Shamans averaged 26.92% more DPS than Frost Mages (+776K DPS).
Frost Mage+0.78% DPS
When upgrading the gear for both specs by +44 ilvl, Frost Mage DPS improved the most, +0.78% more than Elemental Shaman. In total, Frost Mages DPS increased by 67.95% and Elemental Shamans by 67.17%.
This category evaluates the burst DPS potential of each spec by simulating a short encounter with full raid buffs and bloodlust.
This category evaluates the solo DPS potential of each spec by simulating a typical solo delve encounter with no raid buffs.
This category evaluates the pure AoE DPS potential of each spec by simulating both specs to fight intense groups of adds, with nearly 16 adds at any given moment.
This category evaluates the performance of each spec when introducing occasional movement and stuns during boss encounters.
This category evaluates the latency tolerance of each spec by simulating both specs with moderate-to-high latency. This can indicate which spec performs better when facing high latency, whether due to far server/realm location or limited access to quality internet.
This category evaluates the skill level required to play each spec by simulating both specs with greatly reduced player skill. This can indicate which spec is more forgiving of mistakes or has a simpler rotation.
The comparisons above highlight some of the key differences between Elemental Shaman and Frost Mage in the current meta, which can hopefully help you determine which spec to play or group up with.
Elemental Shamanvs Frost Mage
Frost Mage+5.21% DPS 
Elemental Shaman+26.92% DPS