What predators are you trying to stop? Will that wood fence stop them? How big are the openings in the wood fence? In many cases the answer will be that it will work fine but I don't know what that fence looks like.
Are you going to put a roof on your run? I suggest you want to be able to walk in your run without bending over. If you do put a roof the far side of the run may need to be a bit higher so you can comfortably walk in there.
Always good to hear from you Ted but I'll question you on this. I went to Lowe's website and priced a roll of:
100' long 4' high 2x4 welded wire. Cost $108 US
100' long 4' high 1/4" hardware cloth. Cost $160 US.
I don't know about Canada but in the US I don't think cost benefits is why so many people use hardware cloth.
@Faithfulground again what predators are you trying to stop. Properly attached hardware cloth will stop almost any predator though a big dog or big boar raccoon may be able to tear it if it can get a grip. 2x4 welded wire will let snakes, many weasels, rats, and mice through but if it is properly attached should not be torn by anything smaller than a bear.