Cultural traits persist longer than accents. Not forever (if you wait many generation), but longer (often a few generations). If you talked to me or my sibling on the phone, you wouldn't be able to tell I wasn't any other American. On the other hand, many of our personality traits are reflective of the culture our parents were born in.
Growing up, I always thought those were places where I was socially awkward and didn't fit in. It wasn't until I did a deep dive into cultures and started managing international teams that I saw that these lined up completely with the culture my roots come from.
I can't talk about specific differences since how people from India differ is not the same as how people from Tanzania differ. A good survey for general differences is Hofstede's writing.
In addition, there are major differences in communication styles. I will mention a few major ones:
1) How positive one is. Americans always smile. They're always doing "well," "fantastic," or similar. Eggs start at medium, and go up from there. That's not true of most of the world.
2) When and how much one shows emotions or talks about personal details in professional settings. Immigrants from cultures who show them less (e.g. Japan) seem emotionally stilted. People from ones who do this more (including many African American communities -- you can't get more American than that) seem unprofessional.
3) China: Emojis / "cute pictures / etc. in professional communication.
4) When one disagrees (and especially across hierarchies), how, and especially how much confidence one shows. This is a gender difference too.
5) Sense of humor (what's funny -- watch foreign films and see where people laugh)
Most people have no problem getting over the big stuff (e.g. Middle Eastern gender relationships), but it's the subtle stuff that puts one in an uncanny valley. There's an almost fractal expansion of nuance in subtle ways language differs, what's appropriate, etc. That's really tough to manage unless both sides are expert in it.