When you say he's just 20, the meaning usually is that he had his birthday recently. So he's just 20 might mean he had his twentieth birthday last week. However, it would be more usual to say: he's just turned 20.
In the phrase: he's only 20, the underlying meaning is that he's still quite young. For example: he hasn't got a serious job with a pension scheme yet. That's okay, he's only 20. He's got time.