First, the current system is pretty bad. Florida getting the 3rd seed in east with 94 points? Phoenix 3rd in the west with 97? This is dumb, and feels artificial.
The divisions are also jut not interesting in the general standings, which is easily displayed in the fact that standings are normally presented in conference format. I'm betting a lot of casual fans would be hard pressed to name all teams in their division quickly, let alone in any other division. I think getting rid of divisions is easy in this situation.
Then again big conferences do nothing to promote rivalries and not enough to limit travel, and 3 conferences has the downside that it makes planning playoffs tricky. There's just no "natural" way of setting things up, and a lot of ways to end up with tons of travel and lots of games in faraway timezones.
So 4 conferences it is, and I think that has a lot of potential. I'm guessing one of the big things the NHL is looking for is that the fans of each team in the playoffs could see all "meaningful" playoff games of the first two rounds in their timezone or maybe 1 timezone away.
I also think 4 team playoffs to decide 4 conference winners who go on to play another set of 4 team playoffs is easier and more interesting to follow than 8 teams battling for 2 spots, even if there is a lot of repetition with your opponents. (Then again, that would require your team making the playoffs repetitively. I think I might live with that.)
Next I'm going to go into stuff which is very much just my opinion.
Sports is about winning things, and the current setup of the league makes it hard for a team to win anything at all that matters.
As things are, conference banner is either heavily connected to losing the cup finals, or completely overshadowed by winning the cup. Neither situation is good for creating an emotional experience of "this is a thing we won". Also, on average you get one every 15 years, which is a really long time. And most teams will not win them that often.
And the division banners, who remembers fondly the division championship of '06? Which is more talked about, that or not getting out of the first round? Who got bragging rights for '04 season, us or Vancouver?
I think having a cup winner + 3 conference winners would be an improvement.
I'm also sure the conference winner playoffs will have a new name created by NHL marketing, a'la "Frozen Four" to play up that difference between the two stages of the playoffs.
Or maybe they'll just call it the Stanley Cup playoffs