To be kidding is a colloquialism, technically "kidding" means giving birth to a baby goat, a "kid." However, virtual no one uses it in this sense. For all intents and purposes, "joking" and "kidding" are synonyms. I believe different cultures use one more than the other (in Canada, "kidding" seems more common; in South Africa, where in general the langauge is fairly close to UK English, "joking" seemed more common. That's my experience). They mean the same thing.

I'd agree with the other members that there's little difference in meaning. As for the translation, I'd translate both kidding and joking as 'plaisanter' ie saying something that isn't true, just for fun. I'd translate 'faire une blague' as 'play a joke (or a trick) ' on someone, for example a practical joke. 'Taquiner' is 'tease', which is more like poking fun at someone - which isn't necessarily the same as joking or kidding.

"Kidding" is a type of joke - it means lying to someone "Hey, I sold your dog! Aha, just kidding" or teasing them "Hey, you're so ugly Hello Kitty said goodbye to you! Just kidding...".