He ran out of his house(,) with his wife following behind.
Countries all over the world competed this year(,) with Brazil in second place.
Is there are comma needed here (grammatically)? If so, how is this "with" different from others?
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With can imply two different things here:
You can actually drop the with in the second case: he ran out of the house, his wife following behind. If you do that you need the comma really, because the two clauses aren't as obviously separated. When with is there it acts as a connecting word.
Your Brazil example fits the first reading, Brazil in second place isn't a separate thing occurring at the same time as countries all over the world competed, it's directly connected to that idea. It's describing something about how countries competed, it's additional information. So you need the with in this case (you can drop it but that's an informal style) - but that means the comma is optional. It just adds a pause, or a clearer sense that you're adding some extra detail, so it's a style thing really.
That's a lot of information you didn't really ask for, but hopefully it helps make sense of things!
A comma is not required proceeding a preposition (with, of, by, etc).
However, if included, an implied meaning could be "this is extra, information not critical".
Who did he run out of the house with?
He ran out of his house with his wife.
We should not use a comma because this is an answer, which is required for the answer.
What did he do?
He ran out of his house, with his wife following behind.
Here, we answered the question, and then we wanted to say, "oh yeah, and some other information for you that's extra".