There can be no blanket answer to this question as there is quite a difference among available cheeses.
Cheap American cheese (the processed kind) are not made with any kind of milk, and are a based instead on oil. When the cheese melts, it separates and becomes greasy. When it returns to a solid, the texture of the "cheese" is more like plastic than anything good. However this class of cheese also does not have any redeeming value--it's high in sodium, no calcium, and essentially no protein either.
Some cheeses do not fundamentally change when heated past its melting point, such as a good goat/chevre style cheese. These cheeses usually have a fairly low melting point, and retain their properties when they return to solid form. There will be little difference between melted and solid forms of this cheese. Plenty of protein, based on milk, has calcium, as well as the fats. Some cheeses have carb content (sugar), but most don't.
Other cheeses have high melting points, such as a Parmesan cheese. These tend to retain their form very well and have very little change due to heat--even less change than the softer, lower melting point cheeses. These cheeses have plenty of protein and not quite as much fat as the softer cheeses.
Even these guidelines are broad sweeping generalities. You'll find cheeses that live somewhere in between the general classes I listed here. If the starting cheese is bad for you (processed cheese product) the melted variation isn't going to be any better. Heat can be a destructive force, but even cooking eggs doesn't significantly change the protein content of the egg. The same principle applies to cheeses.
Typically in cooking, the structure of the proteins will change somewhat. This is why cooked chicken is white and firm instead of pink and spongy. Due to the relatively high fat content of most cheeses, the protein isn't in high enough concentration to really have a substantial change in configuration. It's unclear to me whether the cooking destroys some proteins or simply reconfigures it at the molecular level. All I know is that there is a change.