All of these are dependent, of course, on the players involved staying healthy and no new acquisitions over the course of the season.
1st Base: Mark Teixeira vs. Lucas Duda. Teix is one of the best players of the last 10 years, and proved until his late August injury that he's back to his old form. Duda is... Lucas Duda. Edge, Yankees, solidly.
2nd Base: Starlin Castro vs. Neil Walker. Personally, I'd rather have Refsnyder here, but the Yankees decided (for the 2nd year in a row) that they wanted the proven star at the position (and letting Stephen Drew be that last year was a mistake). The Mets picked up Walker to take the place of the departed Daniel Murphy, but he won't be able to match Murphy, let alone Castro. Edge: Yankees, solidly.
Shortstop: Didi Gregorius vs. Asdrubal Cabrera. At the start of last season, the Mets would have had this, as Cabrera's had a decent career and Didi was unproven. Not anymore: Didi's already better than any shortstop the Mets have ever had, including the vastly overrated Jose Reyes. And Cabrera, at this point in his career, is no Reyes. Edge: Yankees, solidly.
3rd Base: Chase Headley vs. David Wright. When Derek Jeter retired, people who thought they knew baseball said Wright was the new face of New York baseball. Fast-forward to October, and he was in about 7th place to be the face of the Mets, behind their "aces," Murphy and Cespedes. Granted, Headley's not great, but I know who I'd rather have in a key situation. It's not the proven failure Wright. Edge: Yankees, solidly.
Left Field: Brett Gardner vs. Michael Conforto. Gardner faded down the stretch, but he still added power to his speed and defense. And are the Mets really gonna trust a 23-year-old with 194 major league plate appearances to be a starting outfielder? Says a lot about them. Edge: Yankees, in a landslide.
Center Field: Jacoby Ellsbury vs. Yoenis Cespedes. Say what you want about Ellsbury's health, but he's a proven clutch performer with multiple tools. Cespedes is a one-dimensional player, and that one dimension didn't work out in the postseason. Edge: Yankees, slightly.
Right Field: Carlos Beltran vs. Curtis Granderson. Both of them looked done for most of the season. Beltran still has more run-producing ability. Edge: Yankees, slightly.
Designated Hitter: Alex Rodriguez vs., oh, that's right, the stupid National League doesn't use the DH. Edge: Yankees, in a landslide.
Bench: Dustin Ackley, Slade Heathcott, Aaron Hicks, Rob Refsnyder, Austin Romine, Gary Sanchez and Mason Williams (Greg Bird is out for the season due to injury); vs. Eric Campbell, Alejandro De Aza, Wilmer Flores, Dilson Herrera, Juan Lagares, Brandon Nimmo, Kevin Plawecki, Matt Reynolds and Ruben Tejada. Does anybody on the Met bench excite you? If so, you need to drink less coffee. The fact that the Yankees can call on Refsnyder and Hicks bodes well. Edge: Yankees, slightly.
Catcher: Brian McCann vs. Travis d'Arnaud. d'Arnaud is the reason the R.A. Dickey trade isn't a total wipeout in the Toronto Blue Jays' favor. (Well, that, and Dickey hasn't been so hot for the Jays.) But McCann is a better fielder, a better handler of pitchers, and a better hitter. Edge: Yankees, solidly.
Starting Rotation: Masahiro Tanaka, Michael Pineda, Nathan Eovaldi, CC Sabathia and Luis Severino, with Ivan Nova as the step-in man; vs. Matt Harvey, Jacob deGrom, Noah Syndergaard, Bartolo Colon and Steven Matz, with Zack Wheeler as the step-in man. The Yankees only had 1 postseason game last season, but that was due to the hitting stopping. In contrast, in the biggest games they (Colon excepted) have ever seen, the Mets' starters embarrassed themselves in the World Series. The Yankees hit them hard in their regular-season Interleague games. NOT ONE of the Mets' starters would crack a fully-healthy Yankee rotation. "Aces"? The Mets haven't had a true ace since they let David Cone get away after the 1991 season. Edge: Yankees, solidly.
Bullpen: Vicente Campos, Nate Goody, Jacob Lindgren, Bryan Mitchell, Branden Pinder, Nate Rumbelow, Chasen Shreve, Kirby Yates, Andrew Miller, Dellin Betances, and closing with Aroldis Chapman; vs. Antonio Bastardo, Jerry Blevins, Sean Gilmartin, Rafael Montero, Addison Reed, and closing with Jeurys Familia. Bitch, please, did you even watch the World Series? The Mets played 5 games, led in all of them, and blew leads in all of them (including the one they won anyway). The bullpen is the biggest reason the Mets are takeable, if only a team will step up and do it, as the Yankees and the Royals both did last year. Edge: Yankees, in a landslide.
Manager: Joe Girardi vs. Terry Collins. Each manager has won just 1 Pennant. Girardi doesn't know how to handle a pitching staff, whereas Collins won his Pennant last year despite an atrocious bullpen. This may be the Mets' only true edge. Edge: Mets, slightly.
Divisional Opposition: The Yankees have to get past the Toronto Blue Jays (who had no business being in the postseason), the Baltimore Orioles (no real threat), the Tampa Bay Rays (done now that Dirty Joe Maddon is no longer their manager) and the Boston Red Sox (horrible 2 years in a row, but since they cheat and their bosses have money, count them out at your peril; while the Mets have to get past the Washington Nationals (already established chokers, but signed the Mets' best player from last seas), the Atlanta Braves (some Playoff experience), the Miami Marlins (not very good) and the Philadelphia Phillies (a desiccated husk of a once-great team). It was the weak Division that allowed the Mets into the Playoffs. They can't count on winning it again with themselves diminished and the Nats improved. In contrast, the Yankees' only serious challenger is those pesky Blue Jays, and they haven't improved, while the Yankees have. Edge: Yankees, slightly.
Conclusion: Aside from having a manager who relies on his eyes instead of a damn binder, there is no area where the Mets have an edge over the Yankees. None.
The Mets will be lucky that 2 Wild Card berths are available, because they're not winning the NL East in 2016. The Yankees will win the AL East; how they handle Playoff opposition is another matter.