Personality and values impacts fair few things. To name a few:
What
needs the creature has. Dwarven civs all have same culture, so these are relatively stable.
Whether they will make friends, romance someone and marry them (beyond their orientation, that is). I look for how much they value friendship, romance and family as well as how friendly, obsessive and lustful they are in particular, but this is mere surface interpretation that doesn't consider how well units' personalities match - grudge is obvious, but there's bit of emotional room between a grudge and a lover.
What gives them happy thoughts, sad thoughts or how deeply they're affected (stress vulnerability, sacrifice, anxiety, bravery, valuing/disliking x).
An often-mentioned example is whether dwarf gets happy or bad thought at feeding patients depending on how willing to help others they are.
But this is why some dwarves in particular go nuts. You have dwarves that are perpetually dour, dwarves that get stressed easily and dwarves who get stressed easily and who are perpetually dour.
I don't think it impacts how happy they're learning/improving a particular skill, though it can impact their happiness at learning/improving any skill.
The valuation of sacrifice means they're less likely to feel unhappy about missing something they personally want for instance, iirc.
Preference that matches the job boosts job success rate, i.e. a legendary+5 carpenter producing a ⛭bed⛭ 30% of the time instead of 27%. I'm not 100% certain on whether focus impacts quality in addition to job speed, here.
How long they work without a break,
supposedly.
Not a complete list; with focus, talking and thoughts I don't think there's much that isn't affected by personality in some way or another.