From what I can tell, based on regular season finishes (which always doesn't tell the story), since expansion for the 1967-68 season there have been 38 NHL Finals and the team with the higher regular season point total (including 1975 and 1992 when the teams had the same point total but the eventual winner had more regular season victories) has won 29 Cups and the team with the inferior regular season point total has won 9 (1971, 1974, 1980, 1983, 1985, 1986, 1990, 1995, 1997).
Based solely on regular season points, 1980 (25 points), 1995 (18 points), 1990 (11 points), 1983 (10 points) and 1971 (10 points) saw the largest point differential between the Cup winner and loser.
Based on regular season finishes:
Team with best record has won 17 Cups
Team with second best record has won 6 Cups
Third best record has won 2 Cups.
Fourth best record has won 4 Cups.
Fifth best won 3 Cups
Sixth best won 3 Cups (winners in 1983, 1992 and 1993)
7th best won 2 Cups (1986 and 1991 winners)
8th best has zero
9th best 1 Cup (New Jersey, 1995)