WHY are there two kinds of 'possible choice' questions?

Ordering does matter in the first question. It even says so - "how many different orderings of songs can she choose?". So in the first one, playing "Twinkle Twinkle Little Star" before "Happy Birthday To You" is a different result to playing "Happy Birthday" first.

In the second one, the DJ is choosing groups of records, i.e. he's just pulling them out in any order. So in that case, the set {"Toto I", "Toto II", "Toto III", "Toto IV"} is considered the same set as {"Toto II", "Toto IV", "Toto I", "Toto III"}.

To switch between the two, you can start with the first case (pick 4 things, in order, from a set of 7), then having listed all of those sets, you note that they can be grouped together into equal-sized sets of redundant orderings. For example, let's say we're just picking 2 options out of {A, B, C, D}. So if order does matter, then the options are:

AB
AC
AD
BA
BC
BD
CA
CB
CD
DA
DB
DC

But you can then group those together into options that are re-orderings of each other, i.e.

AB BA
AC CA
AD DA
BC CB
BD DB
CD DC

So you picked $4 \times 3$, but then grouped them into sets of $2 \times 1$, giving you $\frac{(4)(3)}{(2)(1)}$ different options in the end.

Let's compare the two question:

Question $1$:

A DJ is choosing four new records for his collection. He has seven available choices. How many different orderings of songs can she choose

Quesiton $2$:

A DJ is choosing four new records for his collection. He has seven available choices. How many different groups of records can he choose?

The first question is like someone asking the DJ how many possible ordering is there? He cares about the ordering of the songs being played. tell me exactly what is the first song, the second song, the third and the fourth. How many possible answers are there.

The second question is someone just asking what are the songs being played tonight. We just want to know what are the songs, I don't care about the order the songs that they will be played.

The order does matter in the first problem: for the first song, you have $7$ choices, after which you have $6$ choices left for the second song, etc. That is, you are creating an ordered playlist in the first problem.

For the second problem, you are asked to create a 'collection' of songs, and for a collection, order does not matter.

So yes, these two problems are different.