What is the difference between these two theorems?

Question

From Rosen's Discrete Mathematics and Its Applications, 3ed, chapter 6 p. 428-429:

It seems to me that both of them are talking about partitioning a set of n elements into k subsets. It seems like it does not matter if the objects are distinguishable or not. What exactly is different?

Answer 1

You can transfer the formula of theorem 3 to theorem 4 as follows:

In how many ways can you label the $n$ distinguishable objects with $n_1$ labels $1$, $n_2$ labels $2$,$\ldots, n_k$ labels $k$?

Here the label tells in which box the object goes.

So, each arrangement of the $n = n_1+n_2+\cdots+n_k$ labels gives a way of assigning the objects to boxes. But, each permutation of the labels $n_1$, $n_2$, etc. does not change the box an object is assigned to.