Both "italiano" and "d'Italia" mean "belonging / related to Italy", but they are used in different contexts.
We usually say "d'Italia":
- for kings, Re d'Italia (King of Italy), and kingdoms, e.g. Regno d'Italia (Kingdom of Italy) (but we say "Repubblica italiana", not "Repubblica d'Italia", and "Presidente italiano or, more officially, "Presidente della Repubblica italiana", not "Presidente d'Italia")
- for names of competitions: Giro d'Italia (because it's a "giro dell'Italia", a "tour of Italy", not an "Italian tour")
- for awards, with superlatives: "Il miglior ... d'Italia"
Otherwise, we say "italiano".
It's more a matter of context and occurrences than rules. As you get exposed to the language and practise it, you will eventually know which one you should use.
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