If we want to start very simple, keeping our definitions to just two words each, negative liberty means “freedom from,” while positive liberty means “capacity to.”
Another way of thinking about the difference—though again, it’s a rough one—is to see negative liberty as being about the absence of external limits, while positive liberty is about the absence of internal limits.
Typically, libertarians believe the state should only concern itself with negative liberty and should never undertake to actively promote positive liberty. In part this is because we recognize that, in order to give some people the resources they need to get what they want, it must take those resources from others.
.. allowing both negative and positive to claim the label of liberty can make it more difficult to argue against the state actively trying to promote the former at the expense of the latter. After all, who wants to be put in the position of arguing against “liberty?”
In this case, we might be better off saying that only negative liberty is really liberty, while positive liberty ought to be renamed something like “power” or “capacity.”