I am thinking that they were former allies/etc.
I blame Brawl for this - it said that his and Fox's rivalry started at James, yet canonically speaking Wolf is only one year older than Fox.
James' attention sounds plausible. Perhaps Wolf wanted James' attention and affection as well. I could imagine it's possible that Wolf didn't have a father growing up. It's possible James was that father figure. Perhaps Wolf was merely competing for that attention. Maybe Wolf thought that James was too good of a parent for Fox. Maybe as a child Fox was unappreciative of the things James did for Fox and Wolf resented that Fox had everything and didn't care.
I personally like to think that Fox was just a spoiled, selfish brat that got everything handed to him and he never once showed in gratitude towards anyone while Wolf had to work for it. Growing up next to someone (or even just being in the general area) that was like that WOULD probably cause some problems with a relationship.
Another would be that maybe when James was deciding who would be the next leader of Star Fox, James decided to choose Fox over Wolf. Or maybe he was going to let both of them lead Star Fox, and then when Pigma betrayed James, he gave Wolf a house of lies about how Fox was going to run Star Fox alone. I could see how that could push someone to form a team against Star Fox. Seriously, it's always bothered me that Nintendo had an amazing lapse in creativity and just decided to name Wolf's team "Star Wolf." However, having Wolf name the team "Star Wolf" to mock team "Star Fox?" That's a bit comical and just shows how pissed off Wolf is at Fox (possibly childish as well, but hey, it makes the character's a bit more "human" emotion wise).
As it stands there simply isn't enough information given to do anything much make theories however.