Great question, and the answer is: There's no real answer. We would have to go back in time to figure that out, for surely there were reasons one couldn't "mix threads" in those days. Perhaps it was because certain threads tended to rot more quickly in certain environments, thus ruining a whole garment. Or, perhaps it was simply because YHWH said not to mix them.
Generally the halacha follows that wool and linen (flax) or cotton cannot be mixed and there are many ideas why not, but they are all speculation because Torah doesn't actually tell us; this is one of those nebulous areas in Scripture that we can only speculate about.
The train of thought is simply that wool is animal hair, and linen is from flax, a plant fiber, and the two can't be mixed because the animal hair is going to cut through the plant fiber. The whole garment would eventually fall apart, as the material would ultimate disintegrate. It's along the same lines as "new wine, old wineskins (too much expansion too quick from fermentation), and the prohibition against planting different seeds together - i.e., tomatoes with onions, as the tomatoes would take on acid taste of onion.
Today, it's a personal choice if one wishes in our modern times to adhere to the command to not mix threads (because the command didn't have "forever" attached to it, and today most threads are synthetic).
The whole controversy of milk and meat being mixed is tradition. Torah says don't boil a kid in its mother's milk, which implies showing mercy to animals; and in both places in Torah where it is mentioned it also suggests that all the Mitzvot are to be administered with this same kind of mercy and respect that one would show to a relationship between an animal an its mother. In other words, if we are to show this level of compassion to an animal, then how much more compassion are we to show to one another and to the observance of the Mitzvot?
Concerning the issue of boiling a young animal in its mother's milk - this was a common Canaanite ritual offering to pagan gods; indigenous pagan magic practiced by pagan priests to increase fertility and productivity. YHWH wanted to set apart and make a distinction between Israel and the pagan nations around them; therefore He prohibited them to imitate these abominable ways.
All that being said, we don't go about to cause a brother to stumble; those who adhere to this tradition make a very big deal out if it, so we should not be stuffing a cheeseburger in our mouths when we are in their company. However, in today's society where food is imported from all over the world, chances are slim that meat from Argentina and cheese from France came from the same cow....Of course, we also have to remember that if you melt cheese it becomes very difficult to digest and loses its digestive enzymes and healthy properties.
The bottom line is, we must do our best to follow YHWH's commands concerning what constitutes "food" and what God warned us not to eat (i.e., pork, shellfish and other "garbage disposals" of the world). In the above case, He said not to boil a kid in its mother's milk. So, unless you live on a farm and slaughter a baby goat which you plan to boil it in its mother's milk, it in no way suggests we cannot eat a cheeseburger - which wasn't "boiled in milk" in the first place.