In one [post](http://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. 

To avoid starting a chat with a member whose profile page doesn't show that English is his mother tongue, I have to ask native English speakers: 

Is there really a fine difference between the two? If there is, what is it? Is "I did the best I could" a proper phrase to be used at all?