A year-old question, but I just realized that there is another interpretation. The meaning of just that has been addressed in comments and my other answer is "very recently". In that spoken usage, just would typically be an emphasized word.
However, just can also mean "only". In that spoken usage, the word that just modifies ("spoken" and "now" in these examples), would typically be the one that is emphasized. For this usage, the meanings of the examples are different.
John has just spoken to his uncle.
Scenario 1: John and his uncle have been estranged for decades and you hear that they made some form of contact. Did they get together and engage in some activity together, or merely talk by phone?
Scenario 2: You've heard third-hand about information passed on by John from his uncle. You're trying to establish some context to gauge how accurate it might be. Did John meet with his uncle, who actually showed him something, or did they merely have a phone conversation during which his uncle mentioned something that John may have misinterpreted?
The first sentence example would be an answer to these questions. John only spoke with his uncle, there was no other activity.
John spoke to his uncle just now.
Scenario: John has had a number of opportunities to speak to his uncle regarding some matter. Has he procrastinated and broached the subject for the first time moments ago, or has he discussed it on prior occasions, also?
The second sentence example would be an answer to this question.