Jisho.org Forum

I think 済む is more often used to say that you can manage without something (e.g. "I could use a new bag, but I'll manage without for a while."), whereas 済ます is stronger, and (in the negative) would mean that something is absolutely indispensable (e.g. "You really can't do without a car in this town.")

That's just my subjective view, though - if anyone has a different explanation, please post.

There are lots of differences between 済む and 済ます, depending on context. In the situation of -nai de X, i'm not sure about the technical difference. But outside of that situation, there's quite a difference. I take it you simply want information on the -nai de construct, so I won't list the other differences.