In this chapter, the LORD commands a second census and all of the men of the prior census are gone.
This second census is how we can be sure that 40 years have passed. Before now, we were not specifically told that the 40 years were over, but in this chapter we can see that "among these there was not a man of those who were numbered by Moses and Aaron the priest" (v. 64), so the entirety of the past generation have died and the new generation has arisen to conquer the promised land. This is confirmed by their recent arrival at the Jordan river, across from the promised land.
From this, we can see that this particular census has all of the purposes of the last one (military planning primarily, see Num 1 for more on that). We are also told that the land shall be divided proportionally to the size of each tribe, as the larger tribe gets a larger share of land. This gives us another purpose of the census, using the size of each tribe to allocate the promised land. We are also told the land is assigned by lot (i.e. randomization). I think this is meant to emphasize the equality of the tribes and to ensure that the stronger tribes cannot muscle into the better land. That wouldn't be fair because the promised land was given to all the sons of Jacob equally.
In addition, this census is intended to show us that the entire past generation has died. Without counting everyone, it would have been impossible to confirm. In conclusion, we now know that the Israelites are ready to invade the promised land since the LORD's curse of the last generation has been fulfilled.
There is a separate census for the Levites, as was done in Num 1, and probably also for the same reason. This time there isn't any comparison of the Levites and the firstborn, probably because the Israelites already paid the random for their firstborn. Or it's possible they pay a ransom again and we just aren't told. See Num 3 for the previous Levitical census.
In this chapter, we are not given a breakdown by clan for the Levites, only the total number of Levites. The number is virtually unchanged at 23,000 exactly (from 22,273). It's important to mention that since the Levites were not included in the census of those over 20 years old, they were possibly not slain in the wilderness. Verses 63-65 seem to imply that a generation of Levites also died in the wilderness, but Num 14:29 specifically says "from twenty years old and upward", which was not true for the Levitical census. Also, there was no representative of the Levites among the twelve spies, while Num 14:34 says "according to the number of days which you spied out the land...", but the Levites did not spy out the land.
The exclusivity of v. 65 seems to indicate that there were also no survivors of the Levites: "And not a man was left of them, except Caleb the son of Jephunneh and Joshua the son of Nun". On the other hand, we know that Eleazar is still alive, so it's possible that the priests and their servants, the Levites, are excluded from the rest of the nation. The separate census certainly supports this viewpoint.
With all that context, this chapter is relatively straightforward. We can see that the structure of the census is nearly the same, except that we aren't told the twelve leaders who oversee the census. In Num 1:4-15 we are given a set of leaders who oversee the first census. In this census we are not given a set of leaders, but we can guess that Moses probably appointed leaders like last time. Like last time, Reuben is listed as the first tribe and Simeon is the second. In fact, the entire list of tribes is in the same order, except that Manasseh is now listed before Ephraim. I discuss this more below.
Another minor difference between this chapter and the census of Num 1 is that this chapter includes a few reminders. For instance, v. 9-11 reminds us of the rebellion of Dathan and Abiram, sons of Reuben. This rebellion hadn't happened before the first census. Verse 11 tells us that the sons of Korah didn't die, which means that they did not follow their father in the rebellion. This is really surprising in a culture with such a strong emphasis on paternal authority.
Verse 19 reminds us that Er and Onan, sons of Judah, died in Canaan. Verse 33 is an interesting side note that some particular fellow, Zelophehad, has five daughters and no sons. This is not relevant to anything previously, but we will hear about the daughters of Zelophehad in the next chapter. More broadly, this chapter contains small genealogies for each tribe, while Num 1 was simply a list of each tribe and the number of men. So there are lots of small differences, but the overall effect is nearly the same.
What I would like to discuss now is the results of the census, and to compare these numbers to the census of Num 1. The overall population of Israel is virtually the same as before, down about 1,500 out of roughly 600,000. As before, the math is precisely correct so you don't need to double check for mistakes. The individual tribes are also mostly the same: Reuben changed from 46K to 43K, Gad changed from 45K to 40K, Judah changed from 74K to 76K, etc.
The most significant change is Simeon dropped from 59K to 22K, losing two thirds of its total population. We have not seen any event that specifically targeted Simeon, so this drop is difficult to explain. There have been several plagues on Israel (the plague of snakes, the plague of the Num 25, etc), but none of these targeted Simeon in particular. The only interesting reference to Simeon I can find is that in Num 25:14 tells us Zimri was a leader of Simeon. It's possible that Simeon was more substantially involved in the sin of Peor and therefore suffered excessively from the plague that followed. I don't have any other explanation.
There's some other minor variation (Benjamin becomes smaller, etc), but the only other thing I want to discuss is the reversal of Manasseh and Ephraim. Ephraim loses about 8,000 men in this census, dropping from 40K to 32K. Manasseh grows from 32K to 52K, and is the largest gainer of all the tribes at +20K. Their positions in the list of tribes is reversed, which possibly means that Manasseh is now leading the third division. Previously (Num 2:18) Ephraim was leader of the third division, but since the order of the census matched the order of the camps, Manasseh is listed as the leader of the third division in this chapter.
Manasseh is possibly the leader of the third division and gained 20,000 men. Both of these factors indicate a rise in Manasseh's power and a diminishing of Ephraim. I said in my commentary on Num 1 and 2 that Ephraim remains militarily significant, and that is still correct: Ephraim becomes one of the most significant tribes in Israel's future. What is foreshadowed here is the growing significance of Manasseh. The sons of Makir (also written as Machir) and Gilead in particular are notable for their power. It is possible the region of Gilead is named after the man Gilead, son of Makir. In the future, many sons of Makir settle in the region named Gilead. Gilead (as a region) is mentioned going all the way back to Gen 31, which suggests that Makir named his son after the region, and not vice versa.
Either way, the overall pattern of this census is that Israel has largely not changed since the last census. A whole generation has been wiped out, but the nation has not been destroyed and is ready to invade the promised land.