It seems to me that "He put as much effort as he could into his work" expresses a strong sense that the "he" in the sentence put some effort into the work under discussion, and that whatever that amount of effort proved to be, it was "as much as" he was capable of putting into it.
In contrast, "He put effort into his work as much as he could" suggests to me that he did direct some effort toward work, when he could. But this formulation leaves me with the lingering sense that he may sometimes have put his effort elsewhere altogether, even though he tried to direct it toward work as much as he could.
My impression may be purely subjective, but I think that the first wording does a much better job of commending the person who made the effort than the second wording does.
The meaning is roughly understood to be the same. However, the reason it is confusing is that by the meaning of the words alone, it is not. People often abridge their language, even at the cost of clarity. In dialogue, it is done intentionally to indicate some element of the speaker's personality. A low class person might not be particularly careful with words, especially if he is somehow disgruntled or put under pressure by an interrogator he wishes to assuage.
The variant you exemplify is fairly common in fiction. However, properly put, it should probably be one of these few others "I am not/naught/nought/nothing but..." with the word "But" introducing the only alleged exception from being nonexistent. You can see the words "nought but" being used this way in Shakespeare's Sonnet XV or more analogously to your example "I am nought but a dead man" in Chaucer's The Canterbury Tales.
Now my dictionary of choice, the A.D.E.L. 1828 by Noah Webster, does list "Only" as a possible definition but keep in mind, that is only with a footnote to the effect that a nugatory word is indeed omitted. The only sense of the word listed in The Online Etymology Dictionary's entry for "but" is its sense as a sort of exception.
When people introduce themselves, the phrase already implies some amount of humility, since you are nothing other than what you claim to be. This is demonstrated by the somewhat ironic claims of "I am naught but the king's son" and "I am naught but the king's daughter" shown in Chapter VI of A Boy of a Thousand Years Ago by Harriot T. Comstuck. Claiming humbleness expresses the sentiment more clearly but then again, since humility is sometimes a virtue, it might not actually be considered all that humble to claim the trait yourself.
Just on the other hand, is a more literal method of stating you are exactly what you claim and carries fewer direct, albeit sometimes similar, implications about your character.
Citations:
Individual words are linked to definitions from Noah Webster's 1828 American Dictionary of the English Language.
Cambridge Dictionary of American Idioms. S.v. "nothing but." Retrieved from The Free Dictionary by Farlex.
The Online Etymology Dictionary: but
"I am nought but a dead man" Chaucer, Geoffrey. (2013). pp. 34-5. The Canterbury Tales: Being Selections from the Tales of Geoffrey Chaucer Into. London: Forgotten Books. (Original work published 1857)
"I am naught but the king's..." Page 1015 of St. Nicholas: An Illustrated Magazine for Young Folks, Volume 28, Mary Maples Dodge, Scribner & Company, 1901
Best Answer
The important thing to realize here is that 'put on a show/act' aren't different idioms, they are manifestations of the same phrasal verb:
From oxford:
There is a subtle difference. 'Put on a show' is also widely used without the negative connotations, in the more literal sense of organizing something grand. See this.