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  1. I have just finished my homework.
  2. I just finished my homework.

I think there must be a difference in meaning.
Could anyone tell me the difference in meaning sentence 1 and sentence 2?

3

Your examples:

I have just finished my homework.

     Vs

I just finished my homework.

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.

I have just decided to sell my apartment.

I'm on the way to the station. The train has just arrived.

I have just finished my homework.

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.

I just finished my homework.

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:

I 've lost my keys! (The result is that I can't get into my house now).

We use the past simple for past events or actions which have no connection to the present----

I lost my keys yesterday. It was terrible! (Now there is no result. I got new keys yesterday).

For better understanding check

http://www.perfect-english-grammar.com/present-perfect-or-past-simple.html

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I have just finished my homework (BrE, AmE).

I just finished my homework (AmE).

There's no difference in meaning.

The British use the present perfect for recent actions, especially with just, already, and yet. Americans can use either the present perfect or simple past with these words.

  • I'm skeptical of your belief about "the British" here. On what do you base this notion? "The British" (whoever they might be) are as wont to eschew the perfect as we colonials. What you say may have been true three decades ago, but today, there's little to choose. – P. E. Dant Oct 22 '16 at 9:09
  • The distinction in usage (American English v British English) is a known difference. But I can't agree they mean exactly the same, as used in the same dialect. In other words, in AmE there is a difference between I just finished... and I've just finished.... – Alan Carmack Oct 23 '16 at 2:15

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