picky and choosy are the same. picky is more common in USA, I feel. pick and choose are synonyms (選擇,挑選)
fastidious is not used often, but it usually means you're very concerned with details, very neat and clean.
fussy means you "make a fuss"(抱怨) when you don't get your way.
usually we say children are "fussy eaters", but in a formal situation you'd say picky or choosy.
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