Digging deep into a variety of games, you're bound to learn a couple of ways where playing as player two instead of player one actually changes the game in some way. This was common in shmups where the player two ship might have different movement, or playing on that side decreased the difficulty a little.
Fighting games are another example of this occurring, as specific players might be given a tiny advantage in frame data, or a certain player is given priority in a trade of attacks to prevent a double knockout. At this year's Cleveland Gaming Classic, I happened to stumble upon one such advantage in Strata's BloodStorm.
The character Mirage has access to explosive traps and a teleport move, so the CPU likes to use this in tandem in an attempt to mix up attacks from both directions - she drops a mine in front of you and then teleports behind you in an attempt to strike you into the mine. I found that if I turtled into the corner and didn't move, Mirage would fail to teleport behind me, and then be dropped directly onto her own mine.
I did record a short video of this on my phone, but then I tried to recreate this in MAME to get a direct capture for a better online video. No matter what I did, Mirage kept teleporting behind me and successfully executing her mix-up trap. I started to wonder if the strategy was something that got fixed over later revisions, but no matter what version of the game I played in MAME, the strategy worked as intended for Mirage:

^Mirage successfully teleports behind me as player one and is able to execute her trap
I was about to give up on the direct capture and trim out a portion of my mobile video, and that's when it hit me - I noticed at the event, I was playing as the second player against the computer. Going back into MAME, sure enough, I could duplicate this trick with very little effort!

^The game blocks Mirage from teleporting behind me as player two and the game pushes her back onto her own mine - with correct spacing, a CPU Mirage (default setting) will repeat this over and over, and you can defeat her without even throwing out an attack
Player one/two advantages have been documented in a number of fighting games such as Street Fighter Alpha and Fighter's History Dynamite/Karnov's Revenge, and I've seen a "player one advantage" was even present at the release of the relatively new Mortal Kombat (1). I only find it hilarious that I have enough arcade experience that I was able to deduce the weird behavior in BloodStorm was merely because I decided to join the game as player two ... the player two joystick worked better on that cabinet. 😏
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Super Smash Bros.: Lots of stuff, but most notably who wins when grabbing on the same frame (lower # port wins) and stage positioning at round start depending on what plug-in slot your controller is in.
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Sailor Moon SNES: P1 wins same-frame throws and can throw out of backdash. P2 is fully invuln throughout a throw tech until they land, so P2 can get a free combo depending on the throw animation.
