Update:
So this is what I have found/seen. Matthew has posted some new scores that I was waiting for him to do so and it confirms what I was thinking. Here is what I have found. Out of 128 missions, the New 3DS is at a disadvantage in every missions that deals with "Capturing keeps to advance", "All attacks are devastating" and, finally confirmed, all the "Quiz" missions. I am not sure if this will also include "Defeat all enemy missions".
We have already discussed the "KEEP" missions. The Old 3DS spawns enemies faster than the New 3DS. I just went and tried to compete against Matthews newly posted time here.
https://www.twingalaxies.com/showthread.php/210848
I'll compare my attempts to his 1:10. Doing everything he did as he did it, and having a perfect instant spawn kill on the 2nd and 3rd battles, the fastest I can get is 1:11. What I noticed in my attempts vs his are pretty crazy. First thing I noticed was that Matthew's Magic Meter was already starting to drain before he even entered the 3rd fight. My magic meter doesn't even go away until I've already killed the 3rd fighter. Did not expect that. Also, Matthew did NOT have a clean 3rd fight and he was still able to beat my time. I believe he can easily get a 1:09, possibly a 1:08 on a perfect run but obviously I can't test that out.
I think the Old 3DS processes the dialogue boxes slightly faster than the New 3DS. There is nothing else in the runs that could explain how his time is just slightly faster, even with a fight mistake, other than the dialogue boxes disappear quick enough that it gives the Old 3DS the advantage.
So essentially what I'm finding is, the Old 3DS does the simple things faster, and the New 3DS does the longer missions faster. I'll have to wait and see if he can beat any of my non "Quiz" or "Keep" fights to confirm that. I'm just now redoing my "Defeat the boss missions", I think the New 3DS will have the advantage there since the bosses are much larger but we will see. It won't take Matthew long to prove me right or wrong there since the missions are incredibly short.
Not including any Quiz missions which I now know are in favor of the Old 3DS(and possibly Defeat all the Enemies and Defeat all the bosses missions) the Old 3DS has the outright advantage on 42 out of 128 missions on the Adventure Map.
At least I know where not to waste my time until I get an Old 3DS I guess. Matthew, if you could at least attempt the Mission A3 (2 Imprisoned) to either match, beat, or not compete on my time so we can confirm these findings that would be great. Thanks.