First, art from the ancient Near East is of utmost importance in these kinds of conversations, due to the fact that we can be relatively trusting of their renderings of dress, hairstyles, etc. Cf. things like the Black Obelisk of Shalmeneser III, things from the Karnak temple in Egypt, etc.

The best, and most accessible, resource for this kind of thing is King and Stager's Life in Biblical Israel. It's something that pretty much any non-specialist can read but it is also very easy to follow, has lots of great pictures of things pertinent to the discussion, and it's bound really, really nicely.

That said, I'll provide several quotes from them that should be helpful (from their section on clothing, pgs. 265-85:

"Changing clothes was an infrequent occurrence, a luxury practically unknown to commoners."

The biblical text tells us that clothing was used not simply for wearing but for helping to carry things like bread (Ex 12:34)

"The undergarment worn by men next to the skin is the linen or leather ʾēzôr, a wrap-around skirt comparable to a kilt or loincloth, held in place by a belt. It was the basic garment of Israelite soldiers and laborers."

"The kuttōnet...placed over the ʾēzôr, was similar to a tunic. Worn by both men and women, it was an ankle-length garment draped over one shoulder, with medium or long sleeves, and ordinarily made of wool. The Israelites customarily wore the kuttōnet while working, gathering it at the waist with a belt or sash."

"A general term for the male and female outer garment, resembling a cloak, is kĕšût...Reaching just below the knee, it was sometimes simply wrapped around the body; at other times it was draped like a toga over the body and tied by a belt. In either case, it protected the wearer from the cold and rain. It was made from a square piece of cloth and could be decorated with a rather ornate hem for persons of high social standing."

There are several other descriptions of various garments throughout this section--but we really want to know about distinguishing features between male/female dress...

"The apparel of women apparently did not differ much from men's, and the terms are similar. There must have been superficial distinctions, however, since cross-dressing was prohibited according to Deuteronomic law (Deut 22:5). The female símlâ was somewhat longer than the male garment. The woman's outter garment was draped around the body and over the left shoulder; the right shoulder was uncovered. The head and shoulders, and sometimes the face, were draped with a veil (ṣāʿîp) on special occasions...The painting at Beni Hasan portrays three of the Amorite or Canaanite women wearing a piece of colored cloth wrapped around and draped over the left shoulder, with the right shoulder bare...One woman's garment is rounded at the neck. No garment is worn over the tunic, which is colored and sleeveless. on the Lachish Reliefs, the women and girls wear an unadorned garment resembling a cloak, reaching just above the ankle. They also wear mantles pulled over the head like hoods. Over the tunic the women have a long shawl covering the head and shoulders, reaching to the ankles."

So, there you have it. Basically it's not about what you wear, but how you wear it that makes the difference.

And why can't we wear linen and wool together? What did Deuteronomy have against natural fibers?

Since this is AskHistorians, and as historians we have to index a source to its proper historical context, the assumption that Deuteronomy can tell us anything about the year 1300 BCE is incorrect.

Deuteronomy should be read in line with King Josiah's reforms as presented in 2 Kings 22 (Mid-7th Century BCE). In this chapter, a high priest "discovers" the Sefer Torah, the "Book of Instructions," which scholars have noticed is similar to our Deuteronomic text. Despite what the Deuteronomy records as the source of its own discovery, the more likely scenario is that the transformation of Deuteronomy into an authoritative code on par with the Sinaitic revelation did not happen until the early post-exilic era. In other words, it is more correct to ask about the mode of dress, in the Persian Empire and Palestine, in the 6th century BCE, and not 1300 BCE.

TL;DR: Deuteronomy as a code comes into being much later than 1300 BCE. For useful comparanda, one should begin by analyzing the 6th century BCE.

Follow on question, was Hebrew clothing significantly different than non-Jews in that geographical area? I know that other biblical portions were sometimes added to warn Jews/Christians against imbibing in secular activities.