give awayのニュアンスの違い

これはニュアンスの問題ではなく、基本的な文法規則です。

前者は、〇〇に入れるのは代名詞。
後者は、〇〇に入れるのは名詞。
（当然、例外ありますが、少ないはずです。語学は大変ですね。）

Last Christmas, I gave you my heart, but the very next day, you gave it away.
You already knew I was gay? What gave it away?
My stepdaughter asked me to give her away at her wedding.

You should give away your unneeded toys.
They are giving away two tickets to the upcoming concert.
Why do they give away samples on the street corner?

この規則は、他の句動詞にも適用されます。

Pick it up.
Pick up the apple.
Throw it away.
Throw away the empty can.
Make it up.
Make up an excuse.
Check it out.
Check out my new car.