1. It's time to let go of the past
2. It's time to let the past go.

3. Check this paper out.
4. Check out this paper.

Q) "Let go" in #1 and #2 has a different attribute from #3 and #4, not in meaning, right?
"check out" is a combination of a verb and a phrasal particle (sometimes called an adverb or preposition). It is a phrasal verb.

"let go" is a series of two verbs. It is not a phrasal verb even though it may deceive you into thinking it is. It is not subject to the same rules and transformations which are possible for phrasal verbs. In fact, "let go" may exhibit a unique set of grammatical patterns in English.

As it turns out, although "let go of + noun" is standard, there are occasional uses of "let go" + noun. For example, children sometimes say things like "Let go my arm!" "Don't Let Go the Coat" is a song written by Pete Townshend, and the song "Let Go the Past" is by David Dondero. The Christian Bible contains this usage as well:

But he shall let go the living bird out of the city into the open fields, and make an atonement for the house: and it shall be clean.

What do you mean by 'attribute'?
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"of" is added unlike #3 and #4. But meaning is the same, isn't it?
I don't see any relation between #1/2 and #3/4. However, the meanings of #1 and 2 are the same.
 CalifJim's reply was promoted to an answer.
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I wouldn't build rules for English on children's language. They go to school to get rid of their childish mistakes in language. However, let go my arm is a contraction of let go of my arm. There is no distinction. Also, regardless of what the second component is, as long as the primary component is a verb and it functions as a phrasal verb, it is a phrasal verb. You are making a mountain out of a molehill. It is an artificial distinction to try to look smarter than everyone else. That is too common among language teachers. I trained language teachers at a teachers' training college. I know a fair bit about it.

Maybe you don't realize it, but you are answering a question that was asked and answered many years ago.

Why not answer questions that are more recent? That way there's a better chance that the person who asked the question is still participating on our forum and can benefit from your answer.