The terms are not quite interchangeable, though it is pretty natural to think that they are.

The "Cosmopolitan Right" is one of the central conditions that Kant sees as necessary for the establishment of perpetual peace in one of his more famous political works, Perpetual Peace. The other central conditions are (i) the universal establishment of democratic governments or republics and (ii) some sort of "international federation of peoples" that would serve as a sort of formal organization of those republics, while falling short of a one world government, a goal that Kant thought was precluded by social and historical circumstances.

Now, the cosmopolitan right is necessary from two factors. First, Kant thinks that we all, as human beings, have a right to land and resources of the earth. But this is not to say that this right is necessarily universal; to take a trivial example, I don't have the right to seize my neighbors house just because we both have a right to the earth in some abstract sense. One way to take this is that we all have a right to exist and pursue our ends (somewhere) on our planet, but this does not mean that we have a right to every scrap of land on the planet, or even to whatever piece of land we happen to covet. Instead, we all, as a species, have a common right to the land.

Second, national sovereignty is a historical fact. Maybe society could have developed differently, but it didn't, so we need to work within the situation we actually find ourselves in (hence the aforementioned dismissal of a one world republic).

The upshot is thus that we need to find a way to (i) respect our common right to exist and pursue our ends on the earth and (ii) respect national sovereignty. The solution, according to Kant, is that we ought to be able to freely visit and travel to foreign nations, but that we cannot settle on foreign lands without the express permission of the relevant government. So, unrestricted international travel, but local control over immigration.

This, in point of fact, is the specific notion of cosmopolitan right as defined in Perpetual Peace. That is to say, the specific notion of cosmopolitan right at stake in Perpetual Peace corresponds to the condition of universal hospitality; the idea here is that we ought to open our doors, in a neighborly sort of way, to foreign travelers, but that we do not have to let them stay indefinitely. To use a more manageable example, Kant is basically saying that if someone comes over to your house, you should let them visit but that you don't have to let them crash on your couch for the rest of their lives if you don't want to.

Now, the reason that the cosmopolitan right and universal hospitality are not quite the same thing is that the translation between the cosmopolitan right and universal hospitality are contingent on the specific historical situation in which we find ourselves, as embodied in condition (ii). Perhaps, if the situation were different, the two concepts would not be equivalent. But, in the actual world, it turns out that they are.

TL;DR: There is a theoretical difference such that it is possible that the two concepts could come apart, but in the actual world, they don't.

*Edit: I should add that Kant does see the establishment of a one world republic as the ultimate ideal. By "dismissal", I meant to highlight that he does not seem to think that this goal is attainable in the immediate future. So, instead, we ought to aim for the "realistic ideal" of a collection of sovereign republics united in a loose, voluntary confederation.