Why is rhyme scheme in a poem important? I know it makes the lines musical, but why does it matter what kind of rhyme scheme it is?

The rhyme scheme of a poem subconsciously affects the way you perceive the writing, much in the same way that the soundtrack of a movie can make you feel afraid, or the way that certain types of rap can make you feel passionate while others make you feel empowered. Musicality is a very primal instinct that humans feel and relate to, and the musicality of writing is not exempt from the ability to sway readers.
For example, consider the idea of a limerick: it follows a very specific rhyme scheme, technically speaking, “AABBA.” What if you changed the rhyme scheme and made it “ABABA?” Let’s look:
AABBA scheme:

There once was a man from Peru
Who had a big snake in his shoe
He had such a fright
When it gave a bite
So he hopped all the way to the zoo.

ABABA scheme:

There once was a man from Peru
Who was filled with fright
By a snake in his shoe
And it gave him a bite
So he hopped all the way to the zoo.

Both poems tell the same story, but the first one reads with a certain sort of bounce that makes it more childlike, fun, and songlike. That is due to the rhyme scheme.
On the other hand, breaking from a rhyme scheme has the power to jolt the reader out of their reading. Suddenly ceasing to follow a rhyme scheme is like a jumpscare in a movie because the reader/listener will not expect the change. For example, the following stanza does NOT break rhyme:

The moon is round and white,
The sea is filled with light;
The fishes swim and splash,
The boat tips over, crash.

Now read the following stanza where the rhyme scheme DOES break:

The moon is round and white,
The sea is filled with light;
The fishes swim and splash,
The boat tips over, screech, pop.

You can see that the first example was easy to skim over and read because it was predictable; however, good poetry is not usually predictable. The second stanza where the rhyme breaks is characterized by the ending of the last line where the reader is forced to step back for a moment and say, “whoa, I did not expect that.” The critical moment of thought where the reader must reflect on the word choice is very important for successful poetry.
Consider using rhyme schemes carefully within your writing to both achieve specific tones and to place emphasis on certain phrases and ideas by breaking from that rhyme scheme!