I know there are differences between American and British English in this area. So when answering, please specify whether you speak American or British English.
The answer is that "I have just eaten them" is normal in British and I think US usage, but "I just ate them" is not normal in British use, or at any rate wasn't until recently (except in the different sense of mplungjan's answer).
The aspectual difference between the simple past and the present perfect is that the perfect is used for past-with-present-relevance, the simple past for, well, simple past.
So "I have eaten it" has some present relevance - perhaps I can still feel the curry burning in my belly; or somebody has just discovered the cake has gone and wants to know where it is now; or I am in the (present) state of having eaten polar bear at some time in my life. "I ate it" is regarding the event on its own without considering any present relevance - even possibly those same acts of eating the curry, the cake and the polar bear.
In a similar way, some expressions of time encompass the present. "Just" and "just now" do, and so normally do "today" and "this afternoon" (assuming it is still this afternoon). "Yesterday", "once", and "this morning" (if it is no longer morning) do not.
In British usage (more than US), we don't tend to use a present-related expression of time with a simple past, or a non-present-related time with a perfect; if we do the latter, it implies that the relevant time is in fact finished.
So (all judgments with regard to UK usage):
- "I have just eaten it" but not "I just ate it" (in that sense)
- "I saw him yesterday" but not "I have seen him yesterday"
- "I have eaten polar bear" and "I ate polar bear once", but not "I have eaten polar bear once"¹.
"I have seen him today" and "I saw him today" are both acceptable, but have slightly different meanings: "I saw him today" implies that the time within which I might have seen him today is over — for example he has gone away. This is even clearer in negative and interrogative cases: "Have you seen him today?" implies that you might still be able to, while "Did you see him today?" implies that you have missed him.
As I say, the judgments above are for UK English: I am aware that US English is not the same in this regard, but I wouldn't like to specify exactly how.
¹ Actually, I've realised that "I have eaten polar bear once" is acceptable, but with a different meaning, "on exactly one occasion" as opposed to "at some time in the past". "I ate polar bear once" is ambiguous between these, but unless "once" is emphasised, will usually mean "at some time in the past". "I have eaten polar bear once" can only mean "on exactly one occasion".
As a non-native speaker I would say there is a different feel to the two examples
I just ate them sounds like
I simply ate them
As a reply to
How was it possible that all the apples disappeared?
I just ate them, that's all.
whereas the second example sounds more like a
Did you eat the apples I left for you?
Yes, I've just eaten them.
I speak American English. "I just ate them" and "I've just eaten them" are not examples of different spellings in British and American English. "I ate them" is past tense; "I have eaten them" is present perfect tense; "eaten" is a past participle--not a verb. However, I think you're looking for something like this: "I learned that" (American) and "I learnt that" (British). "The British speaker spelt it with a -t, whereas the American spelled it with an -ed." This distinction doesn't hold for all past tense/past participle forms of verbs; "it fit" (American) is spelled "fitted" in Britain.
When American English adds an -ed to the verb stem to form the past tense/past participle British English adds a -t instead. When American English doesn't change the form of the verb, i.e. "fit" (present) vs. "fit" (past), British English adds an -ed to form the past tense "fitted." But also, "light" in American English becomes "lit" in the past tense, whereas in British English it becomes "lighted."
So anyway, I hope this is what you were asking about.