The two are fundamentally different.
1) しないで: signifies intention, like "so as not to..." or "so that (I/you/he) may not...".
2) しなくて: simple connective (no intention) "not doing ...", "instead of doing...", etc.
Examples.
1) Please do not make a noise. うるさいことしないでください。
2) Instead of doing that, do this. それをしなくてこれをしてください。
#1 implies someone's intention - he/she is in control of whether or not to do it.
#2 is a simply connecting two phrases, "instead of doing that, ...". なくて is only there to indicate "not that but...", functioning as a connective.