Maybe a better example:
내일은 잊지 않을 거예요.
It is assured/secured that I will not forget it.
I can't for get it (even if I want to forget it).
In this case, you are telling the listener that it is secured, it doesn't depend on my memory. For example I have taken some special measures to prevent it from happening. (For example, I have told my mother to remind me of it tomorrow morning. So, even if I forget it, my mother will tell me to take it tomorrow)
내일은 잊지 않을 게요.
I will try not to forget it tomorrow. / I hope I will not forget it tomorrow. / I have a strong will not to forget it.
In this case, nothing is secured(it still depends on my memory). I will try not to forget it, but no special measure is taken to prevent it from happening.