In many contexts, the meanings are pretty much the same, but you might hear finished more often than completed in casual conversation. For example:
I've finished my shopping.
She finished the song.
He finished the race.
I could use completed in those sentences – the meaning wouldn't change, but the register might sound off.
The word completed can convey some sense of accomplishment. In the context of a race, it might work when the race is a major achievement:
He completed his first marathon last year.
Homework, though, is not really a major achievement, so I think you'd hear finished more often in casual conversation:
“Joey, where are you going? Did you finish your homework?”
That said, you might see completed in more formal contexts, such as a paper on education, or a course syllabus:
Students must complete six homework assignments during the semester.
Fairly familiar usage, ordered by how recently I ate, with common implications:
"I have finished eating" -- I just now finished eating.
"I am done eating" -- I recently finished eating; I will not eat again soon.
"I have eaten" -- I ate earlier today; I am not hungry now.
Best Answer
Your examples:
I think this question is purely about simple past vs present perfect as well as British English vs American English. That's why, you are saying "there must be differences in meaning between them"
As far as the differences in meaning between these sentences is concerned, there is no real differences in meaning between them. As john Lawler mentions in his answer https://english.stackexchange.com/questions/202122/the-correct-usage-of-past-tense-with-just "Different speakers with different habits and experiences will probably see potential distinction or difference to make but nobody will see or make the same ones". Both of these sentences imply the same meaning but tense is different in them, which here doesn't affect their meanings. A speaker choces the one which sounds best to him/her.
The word "just" is commonly used with present perfect tense(most commonly in British English) to indicate that an action has recently happened and influences the present moment of speaking.
According to Cambridge dictionary, http://dictionary.cambridge.org/grammar/british-grammar/just
''just' as a time Expression'' can mean 'recently or very short time before' or after speaking.
Having said that, In American English it's acceptable to use" just" with simple past as well as with present perfect to express that something recently happened.
However, since the language develops over the time, it's widely increasing the use of 'just' with 'simple past' even in British English.
This might be helpfull http://www.eslhq.com/forums/esl-forums/english-questions/past-simple-just-16200/
Simple past vs present perfect
We use Present perfect for actions which started in the past and are still happening now or for finished actions which have a connection to the present.
----A finished action with a result in the present:
We use the past simple for past events or actions which have no connection to the present----
For better understanding check
http://www.perfect-english-grammar.com/present-perfect-or-past-simple.html