They mean the same thing, but they do not necessarily apply indifferently.
(I am going to write in spanglish, becuase you are a beginner student, so you can understand well).
It can be softer or more serious, (The sorry is of heart but it is applied to varied situations).
- I'm sorry I couldn't be present at the party
- Lo siento, I forgot the book you lent me.
- I can not raise your salary, lo siento.
"DISCULPA" (Apologize, excuse me)
It is more polite, you can use it when you want to express sincere apologies, or as the "excuse me".
- I offer you a disculpa for the delay.
- Disculpa for distrusting you.
- Disculpa, what is the price of this jeans ?.
My mom says that this word has a deeper meaning, and she's right ...
Perdón, shows repentance with all the heart.
- God, perdona my sins.
- Yo te perdoné for having lied to me and hurting my feelings, but here our story ends. (love relationship).
- I perdoné what they did to me to have my heart in peace.
I hope I have explained me well.
If not, ask me.
Ten un buen día.