ずに is more formal/literary than ないで. Other than that, they have the same meaning. It’s not impossible to use ずに in spoken language though.
ず with no に is typically equivalent to なくて, just more formal/literary again. But it can’t do things that have て nuance implications, like implying causal relationships.
And it’s also possible for ず to be like ないで, but I don’t recall more of the details off the top of my head.