Why can't Inuzuka clan's dogs use human language when Kakashi's dogs can? Also, how does Kiba and Akamaru understand each other?
Well the difference is quite simple.
the Inuzuka clan's dogs are simple dogs. They can understand human language very well and are the "sidekicks" of the Inuzuka. Kiba and Akamaru understand each other by "Hand Wave". This is not explained any further as it is prescriptive.
Kakashi's dogs on the other hand are Summoned. They are not "real" dogs but spirits, comparable to Gamabunta, Gamakichi and the other Kuchiose No Jutsu summoned animals.
Like how Vogel612 stated the reason why Kakashi's ninken can speak is cause they are summoned creatures. However, the only ninken part of the Inuzuka clan that can actually speak is Kuromaru, at least as of right now.
Also, the Inuzuka all understand their ninken because of their bonds with them.