Interpreting Wittgenstein, especially the Wittgenstein of the Tractatus is always a fraught affair. There is no single, definitive reading of this text, and whatever one says about it is liable to be contradicted by someone else. So what I offer here is not the answer, but I hope it is, at least, an answer.
You write, following proposition 6.373, "And thus the limits of the world cannot change." This seems to me to be precisely wrong or, more exactly, meaningless. The limits of the world are precisely the point whereof one cannot speak (propositions can only express facts; hence, the limits of my language means the limits of my world; what lies beyond the limits of the world cannot be spoken of---it is beyond what language is capable of saying). Thus, we get to this:
6.41 The sense of the world must lie outside the world.
These limits are outside of the world, and thus outside of the possibility of proposition. They can be shown, but not said, not put into a propositional form, for every proposition concerns the world, that is, concerns facts (e.g., 4.1 "A proposition presents the existence and non-existence of atomic facts." [alternatively Sachverhalte is rendered "states of affairs" rather than "atomic facts"]) and what can be made into a proposition is therefore itself a fact.
The contents of the world, in a certain sense, cannot change; the world is only facts. I do not mean to imply that change is impossible; merely that "the world" for Wittgenstein does not expand and contract: it is always the totality of facts. Yet this says nothing about the limits of the world, just as from inside a windowless room, I can say nothing about what the room is like from the outside. Nevertheless that there are limits can be shown, even if not through propositions (for any limit which can be rendered as a proposition is therefore a fact and therefore a part of the world rather than its limit).
So, then, what is it that might lie beyond the limits of the world? There are, I think, at least three important examples Wittgenstein gives in the Tractatus: logical form (see esp. 4.12), ethics (6.421), and the subject (5.632). It is in the discussion surrounding this last point (i.e. the discussion of solipsism from 5.63-5.641) that we find what might be most relevant for understanding why the world of the happy man might be different from that of the unhappy man. Thus, we find the very pregnant comment in 5.63:
5.63 I am my world. (The microcosm.)
To which he further comments:
5.634 This is connected with the fact that no part of our experience is at the same time
Whatever we see could be other than it is.
Whatever we can describe at all could be other than it is.
There is no a priori order of things.
To offer a fairly bold interpretation of this: the world can appear differently to each subject, even though they are presented with the same collection of facts. The subject is the limit of the world (5.632 again), meaning that the subject is transcendental to the world. The subject is an ordering function for the world; it brings together the facts and shows their connections, yet it is no part of the world (5.633). This is what it means to be a limit of the world: to give the world its shape.
And hence we arrive at an interpretation of the sentence in question: it is not that the world of the happy man is some different set of facts from that of the unhappy man; it is rather that the world has a different shape, a different form, a different connection of facts. There is nothing in the world itself which gives it is sense, rather its sense is given from outside the world. Facts in the world do not by themselves "make" someone happy or unhappy. Rather it is almost the other way round: how the facts appear depends on the sense which lies beyond the limits of the world. It is a different world, but only because its limits have changed (and here its important to note that when Wittgenstein speaks of the world waxing and waning it is always the world as a whole which is capable of this---the facts remain untouched by the limits of the world, yet the connection between them, their order, changes).
What might this look like? Suppose you and I were to go to an old medieval church (and I use this only as an example---I have no idea of your aesthetic tastes). To you, perhaps, the decorations, the scale and placement of the altar, the lectern and the pulpit bespeak a sense of wonder and majesty. To me, perhaps, the very same conditions seem cramped and confined, stale and musty, a building setup for a time and purpose that has long since passed. The same building can appear to us as something completely different---and yet this difference does not lie in its arrangement, in the facts about the placement of the various furnishing, in the measurable properties of the building itself. It is something else. And thus we come to what is perhaps one of the most famous sentences of the Tractatus:
6.52 We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer.