That really depends on context. They are in many cases not interchangeable. In the phrase "blant oss", the word refers to something being a part of a group, and then the word "among" would usually be the correct translation. If you translated it as "with us" instead of "among us", it sounds more like accompanyment, or support, more than actually being part of the group. In this sense, I would rather say "med oss" than "blant oss".
But as I said, it depends on context. In context it could be used like "De beste blant oss" - "The best among us", which could (with some good will) be understood as "The best of us". I would however not have translated it as "The best with us". Maybe as "The best between us", but I'm not so sure about that one.
"Amongst" is generally considered archaic, or even obsolete, in American English. Some older literature uses this word, and so it might seem "formal" to some people, but generally it sounds so old-fashioned that it is considered odd in either spoken or written language. https://en.wiktionary.org/wiki/among