「take out」と「take down」の違いはなんですか？

You may or may not recognize this door.

This is the door of my neighbors who seemed to really love New Years. I wrote about them twice before.

http://upgradeenglish.blogspot.com/2010/02/procrastinate.html
http://upgradeenglish.blogspot.com/2010/05/happy-2010-obviousoblivious.html

They put up a wreath at New Years, and it stayed up until last week.

× That's when it was taken out.
～That's when it was taken off (the door).
◯ That's when it was taken down. (best)

• 外す * pitch away《野球》take off（人を勤務から / 体から衣服・帽子・靴・メガネ・指輪などを / 料理をメニューから / 計画・予定・名簿から～を）〔【反】put on〕(definition from Eijiro on the Web)
• take down【句動】1. 〔上にある物を〕下げる、降ろす 2. 〔結った髪を〕ほどく (definition from Eijiro on the Web)

Now it seems that they have moved away. In place of the New Year's wreath, there's nothing but a temporary lock.

I had gotten used to seeing that wreath on the door. Now, when I take out the garbage late at night, I only see this lock.

• Take out the garbage.（生）ごみを出しなさい。(definition from Eijiro on the Web)

These two phrasal verbs have a lot of other meanings. Take a look at Eijiro for more:

• http://eow.alc.co.jp/take+out/UTF-8/
• http://eow.alc.co.jp/take+down/UTF-8/