The difference is a subtle one but methodologically important. There is a distinction in metaphysics between how the world is and how statements or propositions about the world latch on or describe that world. Looking in depth at the former is a different philosophical investigation (which might be driven by the physical sciences) to the latter (which is more geared towards the cognitive sciences).
A metaphysical state of affairs is a logical relationship between parts of the world. This relationship can be understood in a number of ways - in Platonist and Aristotlean metaphysics, we might say that a particular partakes of a universal. Russell gives a good introduction to this kind of metaphysics in his "Problems of Philosophy"; especially chapter 9, "The World of Universals".
When we say of some proposition, say that the tree outside my window has yellow leaves, that it is the case, what we mean to say is that there is a metaphysical state of affairs such that the parts are related in the relevant ways - that the tree and its leaves (or whatever parts or atoms we take to constitute the tree and leaves) are in the arrangement that we take to amount to the tree having yellow leaves.
Being the case is not a property of propositions, but a property of some aspect of the world that we take the proposition to describe. By contrast, Truth is a property of statements or propositions. It says, of a proposition, that the terms of that proposition in the appropriate relations amount to a correct description of something that is the case. On Russell's view, we might say that Truth is described in terms of language that corresponds to what is the case. Where our talk of something being the case is used to make assertions about the world, our talk of something being true is used to make assertions about other assertions, whether by us or by others.
Often, all that truth amounts to is a reiteration of the proposition, and we might say that because our language and our systematic metaphysics should be as close as possible, the truth predicate is in a sense "transparent". This idea is expressed in a number of different ideas under the philosophical umbrella of Deflationism, which has an SEP article worth reading if you want to look in greater depth at the idea.
But to try to suggest that Deflationism is at least not trivial, consider firstly that we can make generalized assertions about propositions whose content we don't necessarily know ourselves:
Everything that Daniel Dennett says is True
And secondly, consider the truth of sentences in other languages and conceptual schemes; what would it take for a non-German to say that an assertion of "Schnee ist weiß" is true?