Obviously vegetarian on the face of it, but maybe not - there is a tweak I will come to later. Vegans don't eat cheese as it's an animal product. Simple as that.
That's the main difference. Vegetarians don't eat anything that an animal or fish had to die to produce. Vegans go further and don't eat anything that comes FROM an animal - so that mainly strikes out, in addition, eggs and anything dairy.
Now we come to the tweak. To make cheese, you need to make the milk clot and separate into curds and whey, and mostly, cheese is made from the curds. Very often, to make it clot, rennet is added and this is made from calves' stomachs - now do you hear the alarm bells ringing for vegetarians? If the cheese was made this way, calves had to die to make it. So this cheese is not even vegetarian.
It is possible to make vegetable rennet and then you can have truly vegetarian cheese. But this seems to be hard to find and if there is nothing on any packaging or elsewhere to indicate that animal rennet wasn't used, you have to assume that it was and it's not vegetarian cheese. So if you haven't checked that, you're not even a vegetarian.