taru-adjective, adverb taking the 'to' particle
prudently; very carefully; with great caution
Literary or formal adverb describing cautious, deliberate action. Often used with と (to) and verbs like する, 進める, 扱う.
彼はその問題を翼翼と扱った。
He handled the matter very carefully.
新しい計画を翼翼と進めている。
They are proceeding with the new plan with great caution.
慎重 is a common na-adjective meaning 'careful, cautious' in everyday language, while 翼翼 is a literary, somewhat stiff adverb used mainly in writing.
注意深い is a standard i-adjective for 'careful, attentive', whereas 翼翼 emphasizes a more formal, deliberate prudence.
The exact derivation is uncertain; the kanji 翼 (wing) reduplicated may suggest the image of moving with the careful, measured flapping of wings, but this is speculative.