For a definition of WAR you can read it on Baseball-Reference [here] and FanGraphs [here]. Basically it is an all inclusive statistic that gives you the number of wins a player is above what an average, everyday bench player or AAA call up.
While the WAR scores are still going to differ between the two sites the gap will be much closer thanks to both sides sliding to a common number on what is "replacement level." FanGraphs explains it as such:
New unified replacement level is now set at 1,000 WAR per 2,430 Major League games, which is the number of wins available in a 162 game season played by 30 teams. Or, an easier way to put it is that our new replacement level is now equal to a .294 winning percentage, which works out to 47.7 wins over a full season. Conveniently, this number is almost exactly halfway in between our previous replacement level (.265) and Baseball-Reference’s previous replacement level (.320), though the number wasn’t chosen solely as an equal compromise.
While this won't significantly change any statistic over any one year (maybe around 0.3 WAR one way or the other in a given year) over a long career it could have a substantial change. For example some players for FanGraphs will lose 15-20 wins over 11,000 or so career plate appearances.
There are still going to be differences between the two sites because both sides have their own fielding views but offensively they are going to be nearly identical from here on out as Baseball-Referenced noticed using the example of B.J. Upton over the last three years where before the change Baseball-Reference had him at a 7.3 WAR while FanGraphs had him at a 13.9 WAR. After the calculations Baseball-Reference still has him lower with a WAR of 7.3 with FanGraphs having him at a 10.7 but that difference is solely on how they value his defense. When you take a look at his offensive WAR from Baseball-Reference is 10.6 and FanGraphs has his offensive WAR at 10.7 so the only true factor that differentiates the two sites is their defensive value for him.
As far as the Pirates goes before the change Baseball-Reference had Andrew McCutchen with a 7.0 WAR and after the change he was at a 7.2. Not overly significant, but a change nonetheless.
Things can be further explained by reading the links above but I brought out a few examples to hopefully help highlight how significant this change is for the statistics community, even if some still don't believe that WAR is a real thing.