Song-writers (and poets) use what we call poetic license; they don't have to follow normal rules of grammar or conventions of prose writing. Their phrases mean whatever you want them to mean. Don't look for logical explanations.
When you post lyrics to song, you should tell us something about the song and what it is about. This is what "context and background" means when we are discussing lyrics.
In this case, the singer is addressing someone with whom she was once in love. Now he loves another woman, but she still misses the love they had together. The lyrics describe that earlier love. I can't explain it word for word, but in a general way, this is how I understand it:
Nothing compares = Nothing be as good as that early love.
No worries or cares =There were no worries or cares when they were young and in love.
Regrets and mistakes / They are memories made. = The things that she might regret or think of as mistakes have become memories that she treasures.