Major Differences in Style
One of the biggest differences is the existence of easy and convenient out-of-combat healing, where it's generally expected that Pathfinder 2e parties will take the time to Treat Wounds between encounters. This free healing means that without time pressure most encounters begin with the party at full health, and encounters are balanced around this. Other party members also have the option to Refocus and replenish some abilities or Identify items during this time so it's not just the healer patching everyone up.
Another difference is the prevalence of Secret Checks for things like Recalling Knowledge or Seeking; situations where the character rolling wouldn't know how well they did (misremembering a troll's weakness to fire, not finding anything but the room has a deadly trap). In 5e and other editions my experience is that other characters will often attempt those checks as well after knowing that the first character rolled badly.
Spells are also much less broadly useful than they are in 5e, with many of the more narrative-challenging spells like resurrect and teleport protected by the uncommon rarity meaning that the GM would need to grant access in some way first. In addition many spells are less able to fully replace proficiency in a skill, such as with the 2nd-rank knock spell only granting a bonus rather than automatically succeeding against mundane locks in 5e. This also contributes to the much easier ability to run high-level games in Pathfinder 2e.
On the other hand characters are universally able to specialize and cast certain types of larger-scale spells called rituals in Pathfinder 2e using their skills. A 9th-level barbarain who chooses to gain expert proficiency in the Religion skill could attempt to cast that resurrect spell with the help of a pair of secondary casters aiding with Medicine and Society skill checks, though with odds of something going horribly wrong on a critical failure.
Beyond this is another difference in that any character can chose to deeply specialize in any skill, allowing wizards with legendary Athletics to wrestle titans or a cleric with legendary Thievery who could steal the king's worn clothes. While a barbarian or a rogue might be able to specialize further these options are definitively allowed for any character of sufficient level.
One thing that might not qualify depending on your definition of style is the availability of all Pathfinder 2e rules in a free officially-supported reference document site the Archives of Nethys. There are definitely similar sites for 5e but in my experience they lack the fullness of free content available here, even with the smaller rules footprint of 5e (46 books total vs more than 100 1st-party books).
Major Differences in System
Probably the most mentioned difference between the two systems is in action economy, that Pathfinder 2e gives characters 3 actions each turn to spend instead of 5e's action/bonus action/movement. Generally this doesn't change too much but it does allow for characters to make things like 3 attacks a round at all levels of play rather than as an automatic function of class like in 5e, with scaling penalties to balance that option.
This particularly plays into the absence of a generic Opportunity Attack for all characters, with Pathfinder 2e fighters and only particularly martially-inclined characters getting it. Paired with the more open action economy this allows characters to freely move around the battlefield without being punished in most cases and makes many dynamic options more viable.
Building characters in Pathfinder 2e is much more involved than in 5e, with every level granting new abilities in feats spread across their character. Instead of getting basically everything from your race at level 1 like in 5e you instead choose a heritage and ancestry feat from broad lists based on your chosen ancestry, with characters gaining additional and more powerful ancestry feats every 4 levels. And for more combat-oriented abilities each character gets a class feat with every even level, also allowing for things like multi-classing or taking archetypes that fit a character theme outside of class. Finally there are a host of skill feats also granted on even levels that allow for characters to specialize in particular skills and unlock more unusual options for using them.
One interesting option in Pathfinder 2e for a character is the option for versatile heritages, allowing any ancestry like dwarf, leshy, or automaton to also be an aasimar reflecting celestial heritage, and having deep support for that through additional ancestry feat options like gaining a halo or celestial wings.
Proficiency is much broader in Pathfinder 2e, with characters having ranks from Untrained (+0) to Legendary (Level+8) proficiency in everything from skills to spells to attacks and AC. Because of this the numbers in Pathfinder 2e are generally much higher than in 5e, with typical level 20 DCs being around 40 rather than 5e's nearly impossible at 30.
5e's advantage and disadvantage mechanics are very easy to use when compared to the larger number of bonuses and penalties used in Pathfinder 2e, though there is a similar ability for advantage that player characters have access to in Hero Points. This allows for players to re-roll a particular check by expending a point, getting one for free each session of gameplay and additional points over the course of the session for acts of heroism/cleverness rewarded by the GM.
Tied to this are the Degrees of Success, where most checks in the game have 4 outcomes based on whether it was a Critical Failure, Failure, Success, or Critical Success. 5e has some support for this in things like attack rolls on natural 20s, but in Pathfinder 2e you can also achieve a critical success by getting a result 10 higher than the DC. With the heightened proficiency this means that higher-level characters are much more dangerous to lower-level ones as a substantial portion of their abilities and attacks will crit while the inverse is true for those lower-level characters.
These Degrees of Success are also influenced by a trait on many encounter-ending abilities like dominate having the 'incapacitation' trait, which protects higher level characters from getting anticlimactically one-shot by increasing their degree of success by one step on a save. 5e also includes protection against this via Legendary Resistances that powerful enemies often have, though this broader mechanism in Pathfinder 2e can also protect the party from lower-level creatures like ghouls getting lucky with paralysis by only happening on a critically failed fort save roll.
Death and Dying
One difference I've found unusually impactful is the initiative changing mechanics when a character gets knocked out and starts dying.
In 5e the most dangerous initiative to roll is immediately after the boss, as such a character will need to roll a death saving throw at the start of their turn before their allies have a chance to help them. This character then likely loses their turn unless they roll a 20, so even if their allies manage to heal them it'll be the bosses turn again before they can act.
In Pathfinder 2e however such a character would have their initiative changed to immediately before the boss's, guaranteeing the rest of the party a chance to help and that the character will get to take a turn if that help is granted.
It really doesn't seem like a huge difference but removing the connection between relative initiative and rate of dying has saved characters more than a few times in my experience, and it protects against a very not fun combo of getting locked out of initiative by an opponent.
Pathfinder 2e and 5e share fundamental underlying principles but have substantial differences.
Systematically there are many more rules for players and the GM to interact with based on the number of choices in character build and enormous quantity of options to choose from, in addition to the more thorough rules covering scenarios that 5e leaves broadly up to the DM.
Stylistically there are differences in how adventuring parties are expected to act, to take some time resting between encounters and rely on each characters' skills more commonly than resolving the problem by spending a spell slot. Additionally players are more freely able to view and learn the rules without needing to commit much beyond their time.