How do Pathfinder Fighters differ from D&D 3.5 Fighters in the first 6 levels of play? Is the feat structure different? What are the changes regarding combat feats and special rules like grappling? Is it easier or harder for a Fighter to do those kinds of stunts?
Pathfinder Fighters get a similar amount of bonus feats, but most of their special abilities are focused around getting better with armor and particular classes of weapons. With armor, the benefits are reduced armor check penalties and not being inhibited in terms of speed with armor. For weapons, they get a +1 bonus to attack and damage that increases over levels and also adds to their combat maneuvers (more on those in a second) with their chosen weapon class: http://www.d20pfsrd.com/classes/basic-classes/fighter
In Pathfinder, all of the odd combat moves are grouped together as Combat Maneuvers. Unless specified otherwise, all combat maneuvers provoke an attack of opportunity. If the AoO hits, the damage done is a penalty to the character's Combat Maneuver Bonus - Base Attack Bonus + Strength modifer + other modifiers (including most feats like Improved Trip, Improved Bull Rush, etc).
The target number for the Combat Maneuver is the defender's Combat Maneuver Defense - 10 + Base Attack Bonus + Strength modifier + Dexterity modifer + other modifiers.
So the process is a little more streamlined and a successful AoO in response to a Combat Maneuver does not automatically prevent it - it just increases the target number for the attacker. However, it is still somewhat challenging to be successful at a maneuver that a character is not specialized to do if they are hit by the AoO due to the target number being higher.
The Pathfinder Fighter is subtly, but distinctly, worse off than the 3.5 Fighter. Which is a shame, because the 3.5 Fighter is not a strong class by any means.
Pathfinder changed the power levels of the Core classes only slightly
Pathfinder has very little effect on the meta-game of 3.5. None of the Core classes changed all that significantly, with the sole possible exception of the Paladin. The general tropes of 3.5 (spellcasters are strictly better than mundanes, in all but the most skewed of cases) remain strong. Stronger, even, since a lot of the mundanes saw as many losses as they saw gains, while spellcasters continue to have phenomenal power.
For reference, JaronK’s Tier System for Classes is the most widely-recognized ranking of various classes’ optimization potentials.
A variety of informal attempts to extend this to include Pathfinder classes:
The thing is, the Combat Feats are their only class feature (everything else may as well be a racial trait; seriously it's all just minor numerical bonuses), and the Combat Feats SUCK.
The "damage output" you speak of is considerably weaker than what the class was capable of in 3.5, tripping is still limited by size category (can't go over one size above you), tripping now requires Combat Reflexes to even function (since it costs you AoOs now, something it never did before), any of the other combat options will eat your feats dry (even with the PF feat increase), and the class is still irrelevant outside of combat.
They fixed NOTHING about the Fighter. NOTHING!
It's like they went onto WotC, found one of the Fighter "fixes" that just added numbers, and then copy-pasta'ed it. They then took an axe to every single feat the Fighter actually used, and expected it to be balanced out.
Short answer, Rogue tier 4 (top of) >tier 3 (bottom of), Paladin tier 5 > tier 4. All other core classes stay same tier, although Druid is clearly at the bottom of tier 1 now.
I would imagine that most of the classes stay in the same tier.
Fighter is still tier 5. Yeah, it can be a powerful one-trick pony. It could in 3.5 too.
So really, nothing much has changed.
About the 3.5 Fighter
Feats are not special; everyone gets those. They are valuable, which can lead you to taking some levels in Fighter, but class features should be better than bonus feats. Levels of Fighter are only for those who desperately need feats, and cannot afford to lose even 1 BAB.
In 3.5, the Psychic Warrior gets the same bonus feats at 1st and 2nd (plus the option of Psionic feats), as well as a number of Psionic Powers, all for the low cost of 1 BAB. The Cleric, with the right Domains, can get feats faster than the Fighter, and gets a ton of other things besides. Two levels of Monk, which is (after the second level) a much worse class than Fighter, is still better in many cases for the purposes of feats.
Dungeoncrasher is gone
In 3.5, the only reason to go Fighter 6 is the Dungeoncrasher ACF from Dungeonscape. In a pure Pathfinder game, that ACF is not available. This eliminates a lot of the justification for slogging through the earlier levels of Fighter.
Zhentarim Soldier is gone
This ACF, from the Champions of Valor web enhancement, gives a Fighter 9 the ability to Intimidate as a Swift action. That’s pretty good; it’s no longer in Pathfinder.
Everyone has more feats
This makes the Fighter’s trick less valuable. In 3.5, levels of Fighter were only for those who desperately needed certain feats. That is less likely to happen in Pathfinder.
Many of the best Fighter feats are weaker
The biggest one is Improved Trip, but notice how, in general, Pathfinder requires two feats to accomplish what only required one before. This does something to counteract the previous point, but this only applies to certain Fighter feats, not across the board. So it’s almost as if everyone but the Fighter is getting more feats.
Small numerical bonuses
The Pathfinder Fighter does have benefits that the 3.5 Fighter did not get. The overwhelming majority of these are small, numerical bonuses to things the Fighter could already do.
The problem with this is that the Fighter’s problem was never the numbers. The 3.5 Fighter could achieve absolutely absurd numbers for the things he did, if he chose to and knew how. This still did not make him a strong class. The Fighter’s problem is that the things he does are quite narrow, and frequently run into problems that their tricks simply cannot be used on.
Another problem is that the Fighter was far from the only class to see small upgrades. The most powerful classes did also.
Conclusion: The Pathfinder Fighter is slightly, but distinctly, weaker than the 3.5 Fighter
The benefits he gets are small, and weren’t what he needed. Meanwhile, the one thing he kind of had going for him has been devalued by the increased feats that everyone gets, and the value of the feats that he can take has been diminished.
These are really small differences, however. Overall, the Fighter has the same relative level of power as he ever did. That was never good, but it hasn’t changed.
I do not have a great knowledge of all of the archetypes available to the Fighter. There may be some that do improve his lot somewhat. In discussions, however, I have not seen any brought forward as “the fix” to the Fighter’s myriad problems.
At best, you might see some Archetype moving the Fighter up a Tier, as Dungeoncrasher did for the 3.5 Fighter.