I've always liked playing around with regressions of hockey stats on Excel so the advanced stats are great for the convenience of data collection.
That said I think people misinterpret the value of the advanced stats, trying to use them as an actual ranking of teams or players. As a simple score, they are virtually non-usable in any predictive manner. However, they are very useful in determining the styles of players and teams, which can help determine ultimate value only in comparing several stats and real time situations together.
For example, one of my favorite stats is the Offensive Zone Finish %. Players with a high value in this stat are clearly Powerforwards and net crashers who cause goalie stoppages while low % are often players who have good skilled scoring and typically jump off the ice after a long shift in the opponent's zone has been cleared.
Similarly high Offensive zone Start% are the top line offensive types while low start % are the more defensive types. A relatively low % is a common trait shared among Selke nominees, but a player with a slightly lower stat isn't necessarily a better defensive player, as there's a good chance he and his team keep the puck out of his zone properly. Looked at with several comparisons, however, this stat can very useful.
As for the overwhelming interest there is in shot differential among these advanced stats, it is certainly a mistake to think these stats can show a straight forward chance of winning in the playoffs. Having done analysis on various forms of shot differential compared to playoff success, I have found there is definitely a "significant" relationship between the two, but the actual correlation (how much the difference in success is explained by shots) is very slim.
It may be a strange concept to those not familiar with statistical analysis, but just because something has "statistical significance" doesn't mean it has a MAJOR effect, it only means that it does have some effect, regardless of how much.
In saying that, the value of shot differential does make logical sense to a degree, a team with a better differential is more likely to have a strong possession game, something that is valuable in the playoffs. That doesn't mean we can write off Shooting% and Save% as luck just because long term regressions DON'T have statistical significance with Playoff Success, it only means that a strong possession game has been the more consistent trait of the teams that have won in the playoffs. Some years might show significance in other aspects, and while one year might show save% to be significant, another might show PP% to be significant. One year might show rainfall in China to be significant, but obviously examining that with any kind of logic shows that as meaningless.
So while I strongly believe that hockey can be analyzed with advanced statistics, there is no way to plug in an equation and get a perfect playoff bracket. Any statistics have to be looked at with logic and scrutiny, and comparing them to actual events in the game is the only way to understand their value.