In saying "Sorry" and to apologize, you admit that you did something wrong. However, the difference between these two phrases is very subtle but still impactful, depending on the situation. Ultimately, it depends on how heartfelt your remorse is. 

I'm sorry.

If you genuinely regret and feel remorse for what you did, you say sorry. You may also say "I'm sorry" if you wish to express sympathy or pity for someone, show your personal feelings about something, or introduce bad or disappointing news to someone.

I'm so sorry for your loss - your grandmother was a kind woman.

I'm sorry that I hurt your feelings when I said that I wouldn't come to your wedding.

I cannot tell you how sorry I am that your wife has left you and your kids. 

I'm sorry to hear that you lost your job.

I'm sorry to inform you that your application was rejected. 

I apologize.

When you apologize, you formally admit or acknowledge that you did something wrong. The difference to saying sorry is that you actually don't have to feel sorry for your actions when you apologize. An apology is more about taking responsibility for your actions than expressing sympathy or sharing your personal feelings towards someone or something. 

I apologize for the inconvenience and appreciate your understanding.

I apologize to the whole team and take full responsibility for my actions.

The governor apologized for his behavior during his last press conference.

The boy apologized to the girl after his mother scolded him.

You should apologize to him for how you treated him lately.