A greataxe with great weapon feat does 6.75 damage a longsword with dueling 6.5
6.75 is greater than 6.5. The math is not as straightforward as that though because of the effect of masteries, critical hits and accuracy. Cleave mastery conditionally adds another 6.75 for 13.5, nor does it consider crits which again compound the die bonus (more dice more effect of the mastery)
If you assume a straight 60% chance to hit, a +3 strength modifier and 50% chance at possibly cleave and a 5% critical hit the actual numbers are
7.7 damage for the Greataxe with GWF and Cleave and 5.9 for a Longsword with dueling. Now those numbers are going to change with the hit percentages, cleave percentages, and a different critical range, but I think what I put above is reasonable values for those and this will hold across a broad range of numbers.
Adding a Greatsword to the comparison, a Greatsword does 8 damage with GWF vs 6.5 with a longsword. Using the same 60% threshold, 5% critical and +3 strength the number for a
Greatsword with Graze is 8.2 damage vs 5.9 for a longsword and dueling.
And if you are using a rapier instead of a longsword you even deal more damage than the greataxe if you consider the mastery.
Vex on a Rapier is not very useful at all in tier 1. In tier 2 it is a lot more powerful, especially on classes with extra attack. The math here is quite complicated and highly situational though.
I am not going to bother with tier 1. If we are assuming extra attack and we assume 80% of follow up attacks are on the same enemy, going again with a base 60% hit rate and base 5% crit rate and +3 strength here is what you get with extra attack in the first round of combat:
Rapier-Vex (80% chance of attacking same enemy twice, +3 strength, 60% hit, 5% crit):
13.2 damage
Greataxe-Cleave (60% hit, 50% cleave opportunity, 5% crit range, +3 strength):
14.2 damage
Greatsword-Graze (60% hit, 5% crit range, +3 strength):
15.2 damage
Going further in the combat past one round will boost the Rapier slightly, but not by enough to close that gap.
The point here is, there is just really no reason to take with a greataxe the feat. Taking defense is just better since the feat does not add any considerable damage.
If you are using a Greataxe you will get extra damage out of the GWF fighting style, if you are using a Greataxe you will get no extra damage at all out of the Dueling fighting style.
Whether you are using a Greataxe or a Longsword you will get an AC bonus out of defense if you are wearing armor. If you think it is better than GWF or dueling then why not just take it instead of these (and I think it is better than either of these fighting styles generally).
But if you take that feat you are now just worse than the 1 handed dude with a shield and dueling. Since you only deal slightly more damage on a crit else the same and have less armor.
Which PC is better or worse is highly situational and dependent on a whole slew of variables. A character with a Shield will have a higher AC than a character without one all else being equal, but they also have a lot fewer options as they do not have a hand free. They can't use any good ranged weapons effectively, it is more difficult to grapple enemies, take the utilize action or cast spells that require somatic or material components. Add to this you are doing less damage and I would strongly disagree that a character with a shield and dueling is automatically better, as matter of fact I would say the guy who is not using a shield is generally the better PC in combat.
If you are using a shield though there is a very strong case for a Sap weapon and Defense. Disadvantage is more debilitating at higher AC.
I stand by what I said, Great Weapon Fighting is about middle of the pack as far as fighting styles go. So is Dueling. The ones that really need a boost are Protection, Thrown Weapon Fighting and Unarmed Fighting.