Distinguishable vs. indistinguishable plays a role when you have an a priori rule to give relative probabilities to different events (the most simple one being a uniform distribution, that is, they all have the same probability, but it can be different, such as exp(- E / k T) or something).
That is, you have a "bag of possible discrete events", and you know that the *relative* probabilities are given by a property of each event (such as its "energy"). In other words, there is an overall normalization constant to be found.
Well, "distinguishable" versus "indistinguishable" comes down to saying what is a non-redundant way of enumerating all the events in the bag ; in other words, what is a "naming scheme" that doesn't point several times to the "same item" in the bag.
Consider the following: consider your bag to be certain Tsars of Russia. Let us say that we want to assign a "relative probability" to them proportional to the duration of their reign.
Now, I am going to keep this list short, but imagine that we have
{Ivan IV (37 years), Catharina the Great (34 years), Peter I (39 years), Ivan the Terrible (37 years), Feodor III (6 years), Peter the Great (39 years)}.
We could calculate, say, the normalization and the probabilities for each of them. We could calculate the probability of having a "the Great" Tsar. But we would make a mistake, because in fact:
Ivan IV is the same person as Ivan the Terrible and
Peter I is the same person as Peter the Great.
So we simply had a redundant naming scheme, and that messed up when we were adding the probabilities or normalizing the set.
We have the same with "distinguishable or indistinguishable" particles.
If we have the list
{particle 1 at position 1 and particle 2 at position 2 ; particle 1 at position 3 and particle 2 at position 2 ; particle 1 at position 2 and particle 2 at position 1 }
we have a list of possible "events" (physical states).
Now, if particle 1 and particle 2 are distinguishable, then that list is all right. But if particle 1 is indistinguishable from particle 2, then that's not true: the first and the last state are descriptions of the same state which is:
A particle at position 1 and A particle at position 2.
Our "numbering" of particles introduced a redundant naming scheme. So if we count them each individually, we make a mistake.
