Counters that use Sino Korean numbers tend to be counters that measure a unit of time.
For example, the Korean counter months is wol (월). Ex: January is ilwol (일월).
Counters that use Native Korean numbers tend to be counters that measure an amount of time that has passed.
For example, the Native Korean counter for months is dal (달). Ex: one month is han dal (한달). Usually for this counter, it’s used in the sense that # number of times has passed. For example, han dal jinettda (한달 지냈다) means one month passed.
Here’s another example of this Sino Korean and Native Korean distinction. The counter for year in Sino Korean numbers is nyun (년). Ex: 3 years is sam nyun (삼년). The Native Korean word for year is he (해). Ex: 3 years is se he (세해). This is not always the case, but in the case of years, the Sino Korean version is more often used.
I think that it should be mentioned that there are some counters that serve as honorifics. For example, myung (명) and saram (사람) are counters used to refer to people. However, when you’re talking about people of high status you should use bun (분). All three counters for people use the Native Korean numbers.
By the way, the counter for age also has this distinction. However, unlike the example above, the honorific counter for age is in Sino Korean numbers while the humble form is in the Native Korean numbers. The Sino Korean counter is se (세). Ex: 2 years old is ese (이세). While the Native Korean counter for age is sal (살). Ex: 2 years old is du sal (두살).
By the way, remember when I said that bun (분) is a Native Korean honorific counter for people? Well, bun (분) is also a Sino Korean counter that means minutes. Ex: 10 minutes is ship bun (십분). Not to confuse you even more, but the hours are expressed in Native Korean numbers. Ex: 10 o’clock or yul shi (열시). Ok, I promise this will be the last one. Beun (번) is a Native Korean counter indicating frequency or the number of times you do something. Ex: 10 times is yul beun (열번).