Since 16th level is the high-end of where the current Paizo Adventure Paths are going to end up at, I will go ahead and do a hypothetical examination of these 3 classes at said level. This will be done along the lines of what I've typically seen in 3rd edition play taking into account, obviously, only the Pathfinder Beta rules set.
For this comparison's purpose, all that matters is raw attack and damage capability.
"Buff" spells of 9th level are not factored in, as I am presuming that the GM has not (yet) gone completely insane and permitted the characters access to such magic. That having been said, naturally his buddies want Lenny to succeed in killing everything he can, so they layer on the following "buff" spells in the one round they have before letting go of the leash:
Greater Heroism = +4 morale bonus on attack rolls (among others).
Haste = +1 bonus on attack rolls and the all-important extra attack at the best bonus on a full attack action.
Prayer = +1 luck bonus on attack rolls & weapon damage rolls (the other stuff is gravy for this example).
Total bonuses are the extra attack at best bonus, +6 attack, +1 damage.
Lenny the Ranger
Lenny the Ranger is Human.
Baseline ability scores are a modest STR 14, DEX 24 and CON 14. The STR and CON go to 18 (+4 modifier) with magic and his DEX to a 30 (+10 modifier) with magic. Base Attack Bonus +16.
The "first" Lenny the Ranger is an Archery Combat Style character, intended to perforate first and ask questions with the cleric.
Lenny's feats are: Weapon Finesse (Human), Weapon Focus - Longbow (1st level), Point Blank Shot (Ranger 2nd via Combat Style), Rapid Shot (3rd level), Precise Shot (5th level), Manyshot (Ranger 6th), Deadly Aim (7th level), Far Shot (9th level), Shot on the Run (Ranger 10th - 'cause the list is messed up a tad), Improved Precise Shot (11th level), Vital Strike (13th level), and Improved Critical - Longbow (15th level).
(BAB +16 +1 Weapon Focus +10 DEX =) +27/+22/+17/+12 baseline. Buff spells (above) improve these to +33/+33/+28/+23/+18. +4 Composite Longbow (+4 STR rating) and +4 arrows further improve these to +41/+41/+36/+31/+26, doing 1d8+13 Piercing, 19-20/x3. <Averaged damage 17.5 points with an average critical dealing 52.5 points.> At point-blank range the attack sequence is 42/42/37/32/27 dealing 1d8+14 <averaged 18.5 or 3d8+42 = 55.5 on a critical hit>. Against the AC 35 CR 18 opponent that his brothers have been carving their way through, Lenny the Ranger can apply Manyshot in his full attack volley against the one he's set his sights on, firing two arrows on the first attack for a total of six arrows a round. Presuming niether misses nor criticals, Lenny the Ranger (Archer) deals an averaged damage output of 105 points for his six arrows. Compared to the two-handed weapon Barbarian and Fighter, this is strong, but clearly not up to par. If he were to be shooting his highest Favored Enemy, adding 8 points of damage per arrow ramps this up to 153 hit points for the entire volley.
However, Lenny (as with his brothers) has Vital Strike. Cashiering his +27 AB shot tacks an additional 1d8 damage on each of his other four attacks, greatly increasing his damage output per arrow: 2d8+14 points (4d8+42 points on a critical hit) <averaging 23 points per shot or 60 points on a confirmed critical hit>. Instead of 6 arrows averaging 18.5 points per hit, he now fires 5 arrows averaging 23 points for an averaged damage output of 115 points, about a 10% improvement with a significantly improved accuracy rate.
Unfortunately for the targeted foe, it is his most Favored Enemy (whatever it was he selected at 1st level), granting a +8 bonus on weapon damage rolls, totalling 2d8+22 <averaged 31 points> or 4d8+66 on a confirmed critical hit <averaged 84 points> per arrow. Presuming all five arrows are delivered without confirming any criticals nor having any missed shots the sorry critter eats 155 hit points per round.
As one can see, there isn't much apparent damage output difference between the two approaches. However, with the higher number of attack rolls the chances of missed attacks are greater (specifically the lowest bonus ones that Vital Strike and Improved Vital Strike address) far out of proportion to the number of critical hits that would be garnered from and - most importantly - Vital Strike and Improved Vital Strike speeds up game play by eliminating a die roll or two per player per round. And of course it's pretty satisfying to roll more than a single die when dealing out damage.
Note that I have NOT factored in the considerable benefits of the Quarry class ability, which adds a +2 attack bonus and automatic confirmation of threatened critical hits...
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As a Two-Weapon Combat Style Ranger, Lenny's ability scores stand nicely, since he's going to want that Weapon Finesse bonus to attack rolls to count.
FEATS: Weapon Focus - Rapier (Human), Weapon Focus - Kukri (1st level), Two-Weapon Fighting (Ranger 2nd), Quick Draw (3rd level), Double Slice (5th level), Improved Two-Weapon Fighting (Ranger 6th), Two-Weapon Defense (7th level), Improved Critical - Rapier (9th level), Weapon Swap (Ranger 10th), Improved Critical - Kukri (11th level), Greater Two-Weapon Fighting (13th level), Two-Weapon Rend (Ranger 14th) and Vital Strike (15th level).
(16 BAB +1 Weapon Focus +10 DEX +4 enhancement on both weapons +6 buff spells =) +37/+32/+27/+22/+37 baseline attack bonus.
+4 Rapier in the primary hand in two-weapon fighting adjusts to an attack sequence of +35/+35/+30/+25/+20 dealing 1d6+9, 15-20/x2 <average of 13.5 points or 27 points on a confirmed critical hit>. Vital Strike is worded poorly for handling two-weapon fighting, so I will play conservative and presume it applies only to the primary weapon. This adjusts to a sequence of 35/35/30/25 dealing 2d6+9, 15-20/x2 <average of 16 points or 32 points on a confirmed critical hit>.
+4 Kukri in the off-hand in two-weapon fighting with the benefit of Double Slice adjusts to an off-hand attack sequence of +37/+32/+27 dealing 1d4+7, 15-20/x2 <average of 9.5 or 19 points on a confirmed critical hit>.
Once per round Two-Weapon Rend will add 1d10+6 points <averaged 11.5 points>.
Once more Lenny's staring down a CR 18 AC 35 critter. Overall, after factoring in Haste and Vital Strike, the baddy is getting stabbed and sliced a total of 7 times, out of which the critical hits will handily balance out the misses. The rapier deals out an averaged total of 64 points, the kukri deals out an averaged total of 28.5 points and the Two-Weapon Rend tacks another 11.5 points for a combined damage output of 104 points. If all 7 blows are against the same +8 Favored Enemy damage bonus that was hinted at above, the damage output increases by another 56 points to a total of 160 points - plenty respectable indeed.
All in all, a two-weapon fighter is presumably gearing more for the not-always-reliable "energy burst" weapons to ramp up the damage output. Tacking on another 7d6 of damage a round averages an increases of 24.5 points per round, totaling either 128.5 or 184.5 against the +8 favored enemy, per round.