The difference is
- あなたは けがを しなかったの*だから: because you didn't get hurt
- あなたが けがをしたのではない のだから: because it's not that you
didn't get got hurt
And, this nominalization can't be rephrased as こと.
(* It needs のだ form here because how you didn't get hurt is justification for suggestion of not crying, or in general, background information for judgement/conjecture. Edit; In this case, the fact that the listener didn't get hurt is a shared information that the listener himself knows well too, which is regarded as a back ground information for judgement of not being worth crying. It's like difference between 'since' and 'because' here. If the sentence was あなたがけがをしたのではないから そんなに泣かなくてもいいのでしょう？, it means the speaker judges that it's the fact that you didn't get hurt that saves you crying so badly.)
Nominalization means turning a clause into a noun phrase, i.e.
- けがをした: you got hurt → けがをしたの: that you got hurt
And you can treat the whole clause as a noun like the below
- けがをしたのは よくない: That you got hurt is not good
- けがをしたのを 見た: I saw that you got hurt (I saw you get hurt)
Whether you can replace の with こと depends on nature of the verb and the position of the nominalized clause, which is really complicated. For example, けがをしたことはよくない is okay while …したことを見た means a completely different thing i.e. "I considered that you would..".
- 見ることは信じることだ: To see is to believe
If you ignore a minor problem, you can somehow rephrase it as 見るのは信じることだ. Now what if you change the predicate likewise?
Unfortunately, this doesn't work as was expected. Nominalization at the position of the predicate of a sentence is saved for a special usage so called Explanatary の.
- けがをしたのだ: It is that you got hurt
Roughly saying, we use that form when we explain things. In addition, you can express partial negation like this.
- けがをしたのではない: It is not that you got hurt