In one [post](https://ell.stackexchange.com/questions/91457/is-the-word-did-a-verb-or-a-helping-verb-in-i-did-the-best-i-could/91458#91458), a commenter maintained that the phrases "I did my best" and "I did the best I could" don't mean quite the same thing. 


If it is true, what is the fine difference between the two?