The answer is partially a matter of style. The Monk would be much more mobile. The Fighter could enjoy using any magic weapons and armor found while adventuring. What about sheer damage?
Let's assume our Fighter has a starting STR and DEX both 16, and will increment DEX at levels 4, 8, and 12 to qualify for all of the two-weapon feats.
When writing attack bonuses, I'll use green for primary-hand attacks and red for off-hand attacks.
L1 Improved Unarmed Strike, Two-Weapon Fighting: -1/-1 , no damage bonus
BAB +1, then -2/-2 for TWF
L2 Weapon Focus (unarmed): +1/+1 , no damage bonus
BAB +2, then -2/-2 for TWF, then +1 for Weapon Focus
L3 Power Attack: +1/+1 , damage +2
BAB +3, then -2/-2 for TWF, then +1 for Weapon Focus, then -1 for Power Attack
damage +2 from Power Attack
L4 Weapon Specialization (unarmed), Power Attack increments: +1/+1 , damage +6
BAB +4, then -2/-2 for TWF, then +1 for Weapon Focus, then -2 for Power Attack
damage +4 from Power Attack, then +2 from Weapon Specialization
L5 Double Slice, Weapon Training 1 (unarmed): +3/+3 , damage +7
BAB +5, -2/-2 for TWF, +1 for Weapon Focus, -2 for Power Attack, +1 for Weapon Training
damage +4 from Power Attack, +2 from Weapon Specialization, +1 from Weapon Training
Double Slice allows full STR modifier to apply to off-hand damage
L6 Improved Two-Weapon Fighting: +4/+4/-1/-1 , damage +7
BAB +6, -2/-2 for TWF, +1 for Weapon Focus, -2 for Power Attack, +1 for Weapon Training
second primary hand attack at -5 because of BAB +6, second off-hand attack at -5 from ITWF
damage +4 from Power Attack, +2 from Weapon Specialization, +1 from Weapon Training
L7 ?: +5/+5/+0/+0 , damage +7
BAB increments. For feat options at this level, see suggestions below.
L8 Greater Weapon Focus (unarmed), Power Attack increments: +6/+6/+1/+1 , damage +9
BAB +8, -2/-2 TWF, +2 Weapon Focus & GWF, -3 Power Attack, +1 Weapon Training
damage +6 from Power Attack, +2 from Weapon Specialization, +1 from Weapon Training
L9 ?, Weapon Training 2 (unarmed): +8/+8/+3/+3 , damage +10
BAB +9, -2/-2 TWF, +2 Weapon Focus & GWF, -3 Power Attack, +2 Weapon Trainings
damage +6 from Power Attack, +2 from Weapon Specialization, +2 from Weapon Trainings
For feat options at this level, see suggestions below
L10 ?: +9/+9/+4/+4 , damage +10
BAB increments. For feat options at this level, see suggestions below
L11 Two-Weapon Rend: +10/+10/+5/+5/+0 , damage +10
BAB increments. Two-Weapon Rend causes possible per-round extra damage d10 + STR x 1.5
Third primary hand attack at -10 because of BAB +11
L12 Greater Weapon Specialization (unarmed), Power Attack increments: +10/+10/+5/+5/+0 , damage +14
BAB +12, -2/-2 TWF, +2 Weapon Focus & GWF, -4 Power Attack, +2 Weapon Trainings
damage +8 from Power Attack, +4 from Weapon Specializations, +2 from Weapon Trainings
L13 Greater Two-Weapon Fighting, Weapon Training 3 (unarmed): +12/+12/+7/+7/+2/+2 , damage +15
BAB +13, -2/-2 TWF, +2 Weapon Focus & GWF, -4 Power Attack, +3 Weapon Trainings
Third off-hand attack at -10 because of Greater Two-Weapon Fighting
damage +8 from Power Attack, +4 from Weapon Specializations, +3 from Weapon Trainings
(Nothing special in the above plan: I expect it is how many dual wield Fighters advance, with their favored weapon used instead of unarmed combat.)
What could our Fighter do with the three "free" feats above? One option is to learn "wrestling" by taking Enforcer, Combat Expertise, and Improved Trip. Another option is to become an effective archer by using second-tier Weapons Training for bow use and taking the three basic archery feats of Point Blank Shot, Precise Shot, and Rapid Shot.
So, at level 13, how much damage is this Fighter doing on average when all six attacks hit?
12 (d3, 6x) + 18 (STR modifier of 3, 6x) + 90 (damage bonus 15, 6x) + 9 (Two-Weapon Rend) = 129 damage
In comparison, a level 13 Monk has five attacks and only two feats to boost damage: Power Attack and Two-Weapon Rend. With Weapon Focus (unarmed) the Monk attacks at +8/+8/+3/+3/-2 and if all five of these inferior attacks hit still only has an average damage of:
35 (2d6, 5x) + 15 (STR modifier of 3, 5x) + 40 (Power Attack +8, 5x) + 9 (Two-Weapon Rend) = 89 damage
So the fighter does an average of 45% more damage. The Monk's extra movement speed and abilities that help escape harmful situations is useful, and he is a much superior grappler. But I expect most groups of adventurers would prefer to have the Fighter with them, who would finish the combat sooner to keep the entire party safer.
(The Monk could close the gap if he could take the Improved Natural Attack feat, but the online Bestiary errata specifies that the feat does not apply to Unarmed Strikes.)