I don't see any difference in the way the Spanish has been written

Question

Purpose:Te regalo mi pulsera nueva con tal de que me dejes en paz.I [will] give you my new bracelet so that you leave me alone.Condition:Te regalo mi pulsera nueva con tal de que me dejes en paz.I [will] give you my new bracelet as long as you leave me alone.

Answer

Hola Paris,
The first sentence expressing purpose: "con tal de que me dejes en paz" means "in order to be left alone".
The second sentence expressing condition: "con tal de que me dejes en paz" means "with the condition that you leave me alone" (so, if you leave me alone I will give you my bracelet)
So "con tal de que", depending on the context, can express a purpose or a condition.
I hope this clarified it.
Saludos
Inma

Answer

I was also struggling to understand this.  Thanks for clarifying, Inma