First, let's start with their weaknesses:
Zigzagoon is weak to fighting type and that's pretty much all.
Poochyena is weak to fighting and bug type. (there was no fairy type in Emerald)
So, Zigzagoon is weak to 1 type and Poochyena is weak to 2 types. My plan is that I am going to add all of their weaknesses and resistances together at the end and see which one is higher (each weakness subtracts one from the total and each resistance adds one to the total) -1 for Zigzagoon, -2 for Poochyena.
Now for resistances:
Ghost type has no effect on Zigzagoon.
Psychic type has no effect on Poochyena, and Ghost and Dark type are not very effective.
+1 for Zigzagoon, +3 for Poochyena.
Now for their moveset before they evolve: (minus fairy type moves)
The highest power attack move Zigzagoon can learn before it evolves is Headbutt, with a power of 70.
The highest power attack move Poochyena can learn before it evolves is Bite, with a power of 60.
+70 for Zigzagoon, +60 for Poochyena.
Now for their evolved forms:
Weaknesses:
Linoone is weak to fighting type.
Mightyena is weak to both fighting and bug types.
-1 for Zigzagoon, -2 for Poochyena.
Resistances:
Ghost type has no effect on Linoone.
Psychic type has no effect on Mightyena, and Ghost and Dark type are not very effective.
+1 for Zigzagoon, +3 for Poochyena
Movesets:
The highest power attack move Linoone can learn is Double-Edge, with a power of 120, but the user hurts itself a lot. Let's minus 20 for that part.
The highest power attack move Mightyena can learn is Crunch, with a power of 80.
+120, -20 for Zigzagoon, +80 for Poochyena
Total for Zigzagoon: 170
Total for Poochyena: 142
Zigzagoon looks like it it is the stronger one by the total, but I ended up beating Alpha Sapphire with my Mightyena. I also have a Zigzagoon that is doing pretty well in my Y game. So, they are both really good Pokemon. It's up to you to choose which one you want now.
I hope this helped! :)