The difference on a long sword vs battle axe is pretty much a 19-20 x2 critical hit vs a 20 x3 critical hit.
If you are comparing a great sword to a great axe then there is a bit more difference. The damage on the great sword will be more consistent, with most damage being around 7 points. It has a relatively low chance of rolling really high or really low. The great axe on the other hand has equal chance of any number being rolled. This means the damage is all over the place. A great axe has a .00417% of dealing 36 + x3 the damage bonus in damage. A great sword has a .00278% chance of doing 24 +x2 the damage bonus in damage.
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Mechanically, most weapons are pretty balanced and it doesn't really matter too much which specific one you pick:
For one-handed weapons, your typical options are:
Martial options:
1d8 dmg, Crit 19-20/x2 (high dmg, high threat, low crit)
1d8 dmg, Crit 20/x3 (high dmg, low threat, high crit)
1d6 dmg, Crit 18-20/x2 (low dmg, very high threat, low crit)
1d6 dmg, Crit 20/x4 (low dmg, low threat, very high crit)
Exotic options:
1d10 dmg, Crit 19-20/x2 (very high dmg, high threat, low crit)
1d10 dmg, Crit 20/x3 (very high dmg, low threat, high crit)
1d8 dmg, Crit 19-20/x3 (high dmg, high threat, high crit)
In the long run, there is little difference between equivalent swords and axes, but personal preferences on crit rates/amounts will come into play.
Generally speaking, the traditional down-side of axes is that you probably aren't going to find as many of them as you will swords in a typical adventure...
As stated, the difference between axes and swords is pretty negligible when it comes to average damage. 19-20/x2 and 20/x3 weapons (or 18-20/x2 and 20/x4 weapons for crit-fishing) are statistically the same other than a few corner cases where the attack roll needed to hit is within the critical threat range.
The falcata (1d8 S, 19-20/x3) is the best weapon for average damage (with the tongi [1d6 P, 19-20/x3] being almost as good).
The reason high critical ranges are "more valuable" than high multipliers is because of "riders" on critical hits: namely, the critical feats at higher level play (basically, the ones with Critical Focus as a prerequisite) and magus Spellstrike ("the spell effect only deals ×2 damage on a successful critical hit, while the weapon damage uses its own critical modifier"). For riders, you want critical hits to happen more often because the additional effects are usually more significant than "just extra damage."
When looking at the average damage one thing to factor in is how likely is the damage going to be outside this range. The number of dice your roll for damage can have a large impact on this. Rolling multiple smaller dice tend to give you more stable damage, but the chance of rolling extremely high or low is much smaller than if you were using a single die. Take for example the difference between 2d6 and 1d12. The 2d6 is going to result in a range of 5-9 point 66.67% of the time. The 1d12 is going to have a 41.65% chance of it falling into that range. The 2d6 roll has a 16.67% chance of rolling 10 or above compared to the 24.99% of the 1d12. The downside of the 1d12 is that it has a 33.32% chance or rolling less than 5, while the 2d6 has a 16.67 or that. The 2d6 has 0% chance of rolling 1 vs the 8.33% chance of the 1d12.
Weapons with a single high damage dice and larger crit multiple have a greater potential for some really high damage shots, but they also have a greater chance of a really low damage roll. Most of these type weapons are usually two handed so are also benefiting from the 1.5 damage from STR and Power Attack. This helps minimize the low rolls and maximize the high ones. This can often lead to overkill, where you do more damage than is needed. The thing to keep in mind is that you are just as likely to get the incredible critical hit on a meaningless minion as you are the Big Bad Boss.
For more flavor wise a axe would have additional uses in day to day fantasy life such chopping wood and its more likely a random ciziten would
have a use a use a for a axe but its depending on the knights backstory eg if they were a woodsmen before becoming a knight having a axe for survival purposes and emergency combat would be easier than having to lug a heavy sword around with them.
If the character is a dwarf then the dwarven waraxe is would be a good axe for sword and board. I don’t think it would be worth spending a feat for a non-dwarf character.
If you want a big two-handed axe the butchering axe does 3d6 damage and has a x3 crit multiple. While the chances of rolling max damage are even smaller than a great axe the extra dice does boost the average considerably. You will be rolling between 8-13 points of damage about 67.58% of the time. You have about an 83.79% chance of rolling 8 or above and a 50% chance of rolling 11 or more. Your chance of rolling either 14 or more is about 16.17%, which is the same chance as rolling 7 or less. This is an exotic weapon so will require a feat. Your chance of rolling a critical hit and max damage is only .0023% so is likely to never happen, but your chance of rolling a critical hit and doing 8 or more points of damage before the multiplier is about 4.12%.
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If the character is a dwarf then the dwarven waraxe is would be a good axe for sword and board. I don’t think it would be worth spending a feat for a non-dwarf character.
It's basically the axe equivalent of the bastard sword (and even does the same base damage, 1d10). A dwarf wielding one of those as a martial weapon and a human spending their bonus feat on Exotic Melee Weapon come out roughly equal, really.
If I am going to spend a feat, I want more than 2 points of extra damage. Even though a human has an extra feat there are a lot better feats to pick up.
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General 'Big Axe' Options for Medium size characters:
Two-Handed:
Butchering Axe [Exotic, Str Requirement] 3d6 dmg, Crit 20/x3 (base damage king)
Orc Double Axe [Exotic, Double] 1d8/1d8 dmg, Crit 20/x3 (double weapons are kinda silly and generally not very good)
Greataxe [Martial] 1d12 dmg, Crit 20/x3 (standard issue for barbarians)
Bardiche [Martial, Reach] 1d10 dmg, Crit 19-20/x2 (Poleaxe with the advantage of Reach)
Dwarven Waraxe [Martial] 1d10 dmg, Crit 20/x3 (kinda meh as a martial weapon)
One-Handed (which can be wield two-handed):
Dwarven Waraxe [Exotic] 1d10 dmg, Crit 20/x3 (one of the base damage kings for one-handed weapons)
Battleaxe [Martial] 1d8 dmg, Crit 20/x3 (standard equipment)
Tongi [Exotic] 1d6 dmg, Crit 19-20/x3 (low damage but unusually good crit stats)
I'm not going to bother going into light axes and axe gauntlet options, as they don't seem to fit the OP's vision...
Feat wise there isn't much that is axe-exclusive.
For some 'okay that was cool' flavor you could have your knight carry a few throwing axes. Then take the sliding axe throw feat. Allows you to ignore shield AC (at a -2 ranged penalty) and if your target is fleeing or flat footed, you get a free trip attempt.
Not practical by any means, but who builds only practical characters? xD
Two other things to keep in mind with average damage and criticals:
1) The keen weapon ability or the Improved Critical feat are pretty much expected from about 7th or 8th character level to keep pace with expected damage output (unless the character has a source of additional damage not multiplied on a critical, such as Sneak Attack);
2) Aggregate bonuses to damage (from Str, enhancement, morale [Inspire Courage bardic performance], etc.) become more important than the base weapon damage dice at higher levels.
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Ok Another Question Any Good weapon abilities beside just strait +s to it.
I can't imagine anything that would Trigger on critical but is there just some Genericly good weapon Abilties
Keen is, of course, quite good if you aren't taking the Improved Critical feat. There are many other good ones, but most of them are at least slightly dependent on your build.
There are actually quite a few weapon qualities that trigger on a critical hit. Everything from the elemental burst abilities (flaming burst, shocking burst, etc.) to exhausting, up to Vorpal. Are any of them worth it? Eh, probably not except in specialized builds.
For the general "which weapon?" question: after 13 years of doing theorycrafting and a ton of actually playing the game I've come to the conclusion that unless you are playing a Vital Strike build (which wants the biggest weapon dice possible) or a build that needs to crit (swashbuckler to regain panache, using critical feats, etc.) you can really divide weapons into two categories. Low damage (20 x2) and good damage (everything else).
Math shows that some weapons do more damage than others. And it is true. But when you are in an actual campaign the difference between a 1d6 (19-20 x2) and 1d8 (20 x3) is minuscule. Because what really matters is "how often does that ever so slightly better weapon end the fight earlier?" The answer is "very rarely." Use the fun weapon.