I find the systems engineering definition useful here. A system is in a certain "state" at a point in time when the system meets a pre-defined set of criterion, i.e. on shopping website, a user's shopping cart enters a different state when the customer adds an item to it.
A "status", in the context of states of a system, describes the transition into a state. For example, the user of our online shop may attempt to add an item to their cart. The transition into the state that describes the item being in the cart may be blocked if the status of the transition signals an error (i.e. the item is out of stock).
In systems engineering, these two words have very specific meanings. From the definitions above, it could also be inferred that a state exists (or does not exist) irrespective of time. A status, however, describes the outcome of an action at a particular point in time. The difference here, I think, holds true even outside the context of systems engineering.
In the context that you describe, the definitions above are still relevant. The state of the project maybe described as, "component A is complete but component B is not; the project is 60% complete overall". This description of the state of the project would be the same at any point in time given unchanged progress of components A and B and unchanged outside influences. The status of the project however would almost certainly change with time, likely with respect to a deadline.