Yes, it's kind of same. 心配しないで can be used as a phrase, like you say " Don't worry"
心配しなくても needs add some sentences and it can't be used on it's own.
Example:
心配しなくてもいいよ。
You don't need to worry about it.
心配しなくても来るよ。
No need to worry, he/she/somebody will come.
However you can use 心配しないで+ sentences but it sounds like casual.
Example/same meaning
心配しないでいいよ。
心配しないでも来るよ。
Just in my opinion and feelings 😌