You can either calculate what happens if the same gem is added to the other or if the gem is removed from both. The results will be slightly different if the attack speeds differ as damage gems are more effective on faster weapons.
If you wanted to add the gem to the axe in your example, take the damage added by the gem (11), multiply it by the attacks/second on the unsocketed weapon, (11 × 1.00) to get the DPS boost (11). Add this to the DPS number on the unsocketed weapon to compare (89.5 + 11 = 100.5 < 103) ∴ the axe does more less with the same gem).
To remove the gem mathematically, you would instead be multiplying the gem's average damage by the socketed weapon's attack/second, then subtract that number from the weapon.
If you're trying to compare just their attack damage (useful for casters that don't care about DPS as much as weapon "swing" damage), divide the end result on either weapon by the attacks per second on said weapon.