Let me add some more examples.
He has few friends. => The number of his friends is very little, he almost has no friends (not enough).
He has a few friends. => The number of his friends is little but "enough", he has several friends.
Concerning Pieanne's examples, I understand them in this way.
Although it was raining, a few people showed up at the burial. => It was "enough" people. That's why the sense is positive.
Because it was raining, few people showed up at the burial. => It was "too few", "not enough" people. There should have been more of them. That's why the sense is negative.