This comparison evaluates the performance of Balance Druids and Shadow Priests, 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
Shadow Priest+2.62% DPS
When simulating for maximum theoretical single-target DPS, Shadow Priests averaged 2.62% more DPS than Balance Druids (+40K DPS).
Balance Druid+21.39% DPS
When simulating for light AoE DPS potential, with small waves of adds, Balance Druids averaged 21.39% more DPS than Shadow Priests (+412K DPS).
Shadow Priest+0.32% DPS
When upgrading the gear for both specs by +44 ilvl, Shadow Priest DPS improved the most, +0.32% more than Balance Druid. In total, Shadow Priests DPS increased by 66.77% and Balance Druids by 66.45%.
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 Balance Druid and Shadow Priest in the current meta, which can hopefully help you determine which spec to play or group up with.
Balance Druidvs Shadow Priest
Shadow Priest+2.62% DPS 
Balance Druid+21.39% DPS