Why does fruit retain its shape when heated in sugared water but break down when heated with just plain water?
It is a case of osmotic pressure. If you heat it in plain water, the sugars inside the cell want to cross the cell membrane and try to equalize their concentration inside and out. With sugar in the water, the osmotic pressure is already equal so they don't need to leave the fruit.
Michael's answer is somewhat correct, but incomplete. Here's a relevant passage from a major food science book:
Sugar addition: Sugar may be utilized in cooking. It offers flavor and some preservation. When large amounts of sugar (amounts greater than that found naturally in fruits) are added to the cooking water at the beginning of cooking, the tenderization is diminished and the shape will be maintained. This is because the water moves out, and the higher concentration of sugar outside of the piece of fruit moves in by diffusion. As well, the sugar interferes with plant pectin solubility. It also dehydrates cellulose and hemicellulose resulting in shrunken, tough walls.
In other words, it isn't only the osmotic pressure keeping the juices inside, but the excess sugar being driven into the fruit.
But perhaps even more notable is the effect on pectin and the hemicelluloses. Pectin degradation is one of the most important things responsible for softening of fruit when it becomes overripe, for example. When cooked, pectin will also begin to break down and eventually dissolve. Sugar solutions slow this process, thereby keeping the fruit more firm. Other elements of the cell walls are similarly strengthened in the presence of sugar.
When fruit is heated in a plain water solution, the opposite happens: pectins become quickly soluble and cellulose is softened. Water also moves into the fruit via osmosis, and sugar diffuses out -- thus actually making the fruit more "plump" (e.g., think of what happens when raisins are soaked).
Note that timing is critical for adding sugar. Osmosis can be reversed (as with dried fruit like raisins), but the breakdown of pectin cannot. Thus, when cooking fruit meant to be turned into sauce, it is often common to cook down in water first (to dissolve the pectin and soften the cell walls), then add sugar after the fruit has already softened, which will drive sugar back into the remaining pulp and produce a thick, tasty sauce.
I actually don't know the answer to this question, but I have a guess based on what I know about candying.
When you soak fruit in sugar, you are replacing the water in the fruit, which helps to preserve the fruit. In contrast, when you cook fruit in just plain water, you are breaking down the fruit. So in one process you are preserving the fruit's internal structure and in the other you are destroying it.
I'm guessing that this process of preserving or destroying the internal structure is what accounts for the lose or preservation of shape. There could, however, be some other process that I'm not aware of.