If we were talking brutal 1, it would be close. But the execution axe has brutal 2, which means you can reroll 1s and 2s on the damage dice. Not only that, but it has high crit, which means you get an additional 1d12 damage on a crit that you get to reroll if you get a 1 or a 2! It's not by much, but according to my math, the axe beats out the blade. Let's assume a +4 attack bonus without proficiency (thus a +4 damage bonus for the modifier) against 16 AC, so the average damages are:
d20---------Fullblade---------Execution Axe
1---------------0------------------0
2---------------0------------------0
3---------------0------------------0
4---------------0------------------0
5---------------0------------------0
6---------------0------------------0
7---------------0------------------0
8---------------0------------------0
9-------------10.5-----------------0
10------------10.5---------------11.5
11------------10.5---------------11.5
12------------10.5---------------11.5
13------------10.5---------------11.5
14------------10.5---------------11.5
15------------10.5---------------11.5
16------------10.5---------------11.5
17------------10.5---------------11.5
18------------10.5---------------11.5
19------------10.5---------------11.5
20------------26.5---------------27.5
Which brings the average damage for the fullblade to 7.1 and the overall average damage for the execution axe to a whopping 7.125. There's not much of a difference there. However, keep in mind that these averages change as the target's AC goes up and down! If you're facing a target with AC 17 or more, the fullblade will actually be the better choice! But if the AC goes 16 or lower, the axe remains champ. The test I did was at the closest they were.
Basically, don't try to pick which blade is better overall, think about what enemies you want to attack! Do you pick off the puny minions while your teammates knock out the big boss? Take the axe, and you'll be good against minions, brutes, and artillery. It's good to have nice mobility options with that axe. Would you rather be the one on the front lines, attacking the strongest thing you see? Then the blade is for you, as it is better against soldiers, elites, and solos (unless your DM changed the defense bonuses with the DMG2) with their higher AC.
It all depends on your own gaming style. Clever how they did that, huh?