Adam Parker of Bring the Books has an interesting take on why the Israelites were commanded not to use mixed threads in their clothing:
This view says that the passage is not prohibiting mixed threads because it was the clothing of prostitutes. Nor is this prohibition merely meant to be a picture of holiness and Israel's distinction from among the nations. Rather, this prohibition had a very practical purpose. Numbers 16:1-40 records an incident when the laity sought to take priestly duties for themselves. In this view Deut. 22:11 (and Lev. 19:19) actually address a real and pressing issue: namely the temptation for the laity to resent or break down the distinction between priests and laity among the Israelites. Given this understanding of the prohibition of mixed threads, we see that God is placing barriers between the people and the Levites to keep such events as the rebellion of Korah from taking place. It is also easy to explain to the skeptic why Christians no longer observe this prohibition. Since the New Testament no longer distinguishes elders from the laity by clothing this command regarding mixed threads is no longer relevant except perhaps in terms of a persistent recognition that the Church still has leaders and elders whom the members are to submit to (Hebrews 13:17).
So rather than the command being about a separation between the Israelites and other nations, Parker says it’s about making a distinction between the priests and the laity. This sounds reasonable, especially since there was a similar prohibition against the laity using the particular blend of incense that was only to be used in the tabernacle (later, the temple) by the priests:
Then the Lord said to Moses, “Take for yourself spices, stacte and onycha and galbanum, spices with pure frankincense; there shall be an equal part of each. With it you shall make incense, a perfume, the work of a perfumer, salted, pure, and holy. You shall beat some of it very fine, and put part of it before the testimony in the tent of meeting where I will meet with you; it shall be most holy to you. The incense which you shall make, you shall not make in the same proportions for yourselves; it shall be holy to you for the Lord. Whoever shall make any like it, to use as perfume, shall be cut off from his people” (Exodus 30:34-38).
But I don’t think this prohibition against mixed fibers for the laity need be merely a practical command meant to maintain the proper, ordained authority structure (and I think Parker might agree with me on this). When creating the Israelite culture through the Law, God used many visual parables to illustrate the setting apart of the holy from the profane. Those who were ordained to stand before God for the sake of the people were set apart in a special way because God is holy, great, and beautiful. All the commands for the specific clothing, the decorations, and the objects in the tabernacle were meant to express this truth about God.
You shall make holy garments for Aaron your brother [the priest], for glory and for beauty (Exodus 28:2).
You shall also consecrate [the items in the tabernacle], that they may be most holy; whatever touches them shall be holy (Exodus 30:29).
[Those not ordained to do so] shall not go in to see the holy objects even for a moment, or they will die (Numbers 4:20).
The separation between man and a holy God wasn’t just explained to the Israelites, it was experienced and seen as they lived out the commands of the Law every day. God spent more than a thousand years preparing us for Christ in this way. Try to read Hebrews 7-10 with the eyes of those who lived the holiness code in the Law—those who were always at a distance from God because of their sin, those who were required to participate in the endless cleansing rituals reminding them of this distance—and you will be awestruck at what it means to be united to Christ, our priest, who has truly and finally made us holy, tearing down the curtain separating us from the most holy place where God dwells.
And Jesus cried out again with a loud voice, and yielded up His spirit. And behold, the veil of the temple was torn in two from top to bottom; and the earth shook and the rocks were split (Matthew 27:50-51).
Parker goes on to explain why he finds this view of the mixed fiber commandment compelling:
a) It has ancient pedigree….
b) It accounts for the previous command for the priests to wear mixed threads….
c) It allows for harmonization within the Pentateuch rather than disharmony.
You can read his full post here.