In the US, "belch" is a polite way of saying "burp" and for this reason "burp" is far more common. Even in formal settings "burp" would probably still be used over "belch" (if it is even talked about). "Belch" is more old fashioned.
To clarify, since you asked how to say it in the US: In the US, I think most people would say "Excuse me" after a burp, but "Pardon me" is perfectly fine. Also, "windy pops" is not a phrase in the US, we would just say the baby burped.
Since belching or burping is not something you usually talk about at the dinner table (), you don't often hear either of those words. We just say "excuse me."
You are more likely to hear belching/burping in a conversation about babies because a belch/burp from a baby is often congratulated. However, I've never heard baby belches referred to as "windy pops," so I think that term is only used in BrE.
The more mature a person, the less we speak the word, "belch." "Burp" isn't offensive, but it's not a word said all the time, though I have no problem using it, especially in a humorous way. I used it just last night, as a matter of fact.
Among junior high school boys, however, - and maybe even younger now - "belching" well, loud, and long is almost considered a "rite of passage" through that stage of growing.
"Boys will be boys," you know. And these days, sometimes even the young girls, too.
And I agree totally with Packard on the distinction between the two.