Didn't they compete for the resources in OTL?
Yes they did, and they've been competing for resources in Africa largely after 1870.
Also how might the rule of the colonies differ from OTL? As I understand it the 3rd Republic tried to federalize the colonies under direct control of the French government. Might the 2nd Empire take a different approach such as the British model of giving each colony its own government?
I would say maybe, maybe not. Napoleon did envision for the creation of an Arab Kingdom in what is now the Interior of Algeria, though nothing came out of it. It might have also been a bit of a pipe dream as well that might or might not be exploited by his son and successor when Napoleon III dies (presumably around the mid 1870's) Whether or not he would federalize the colonies under French governmental control or have a Governor-General in the lieu of Great Britain is up in the air.
Also lets take into consideration the changes that could occur if the 2nd Empire becomes a constitutional monarchy and French individuals involved in the colonies take on similar positions in OTL.
The Second Empire was technically a Constitutional Monarchy throughout it's OTL life-span, (though it wasn't in practice until late into Napoleon III's reign)
Also how might Frances colonial plans in Africa change. As I understand it the French wanted to control north Africa with an uninterrupted boarder from the Pacific to Indian oceans. But that plan fell into conflict with Britain's plan for an uninterrupted boarder from the Cape to the Mediterranean.
I don't think France's plans would change really, and we could see a clash between the initiatives of France against the Cape to Cairo plan that was made by Cecil Rhodes. At least that is my opinion, I'm not really to sure about it.