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QUICK ANSWER
"I'm fine" is a phrase which is often translated as "estoy bien", and "no, thanks" is an interjection which is often translated as "no, gracias". Learn more about the difference between "I'm fine" and "no, thanks" below.
A phrase is a group of words commonly used together (e.g., once upon a time).
1. (general)
An interjection is a short utterance that expresses emotion, hesitation, or protest (e.g., Wow!).
1. (general)
a. no, gracias
Do you want to finish the garlic bread? - No, thanks.¿Quieres comer lo que queda del pan de ajo? - No, gracias.