The Slippery Slope: Once a well-defined principle is set aside in order to permit a controversial policy, there will be no principled grounds to prevent another policy that is deemed unacceptable.
The Camel's Nose: If we allow the group proposing the current policy to have a victory in the current debate, it is inevitable that it will pursue and gain other victories that are currently deemed unacceptable.
I'd put the distinction this way: Being on a slippery slope is objectionable per se, requiring as it does the setting aside of a well-defined principle; allowing the camel's nose under the tent is objectionable per effectum, since it will [almost] certainly lead to welcoming the entire camel. In the former case, you have no intellectual grounds for stopping more bad things from happening; in the latter case, you have no practical means for stopping them.
I don't quite see the tribalism John sees in camel's nose arguments, but I think the distinction is worth keeping in mind, since it suggests distinct responses to the arguments. When you run into a slippery slope argument, you may want to ensure the arguer recognizes that the proposed action is bad even if nothing worse comes along, while to someone offering a camel's nose argument you might ask whether the immediate good is worth the effort to avoid the future bad.