大切 and 大事: valued people and important matters
Compare overlapping uses for value and importance, and distinguish another reading of 大事 from the expression お大事に.
Valued people and possessions
これは母からもらった大切な時計です。 identifies a treasured watch received from the speaker's mother. 大切, read たいせつ, expresses valuing or caring about it.
大事な写真を箱にしまいました。 says that someone put valued photographs away in a box. 大事, read だいじ, also applies to cherished objects, so the two words are not divided by object type.
How important matters are treated
明日は大事な会議があります。 says that there is an important meeting tomorrow. It evaluates significance or impact rather than simply the number attending.
道具を大切に使ってください。 asks someone to use the tools with care. 大切に describes careful treatment rather than prohibiting use.
A well-wishing phrase and another reading
お大事に is a phrase wishing someone, such as an ill person, good care and recovery. 大事 is read だいじ here, not merely an evaluation of an object's importance.
大事に至らなかった can use the reading おおごと to mean that things did not develop into a serious matter. The detail link here is the だいじ entry for importance or value.
Does 大切に使ってください prohibit use?
Show the answer and why
No. It asks for careful, considerate use. It concerns how to use something rather than forbidding it.