"Feliz" and "contento" don't always mean the same thing; they are partial synonyms, but not perfect synonyms. Native speakers may be better guides to contexts where either one sounds good and they mean the same thing and also to contexts where only one of them sounds good or where they do not mean the same thing.
For example, suppose you are eating at a restaurant, and your server comes to offer you more water; you might refuse the offer by saying "gracias, estoy contento" (= "thank you, I'm happy/satisfied = I've had enough").
In this context, saying "gracias, estoy feliz" may come across as saying "thank you, I feel happy now", and your waiter may not know whether to pour more water in your glass or not.
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