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To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\mathfrak{so}(3)$. Now, $SU(2)$ also happens to be a double cover of $SO(3)$, but this seems coincidental/unimportant. It is precisely the universal cover properties which, together with some Lie theory and Bargmann's theorem, lead to a classification of the irreducible projective representations of $SO(3)$.

So, what is the physical importance of $SU(2)$ being the double cover of $SO(3)$? I prefer an answer in terms of the relevant mathematics as alluded to in the comments below this post.

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    $\begingroup$ As stated this is a question exclusively about math, so should be asked on that SE. Are you wondering about physical importance of the double cover? $\endgroup$
    – Marius Ladegård Meyer
    Commented Jun 30 at 5:21
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    $\begingroup$ To add to my comment above, it seems like people often (perhaps wrongly) cite $SU(2)$ being the double cover of $SO(3)$ as being responsible for consequences that are really consequences of $SU(2)$ being the (isomorphic to) universal cover of $SO(3)$. $\endgroup$
    – Silly Goose
    Commented Jun 30 at 5:34
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    $\begingroup$ I see what you mean. I am interested in the physical importance, so I will clarify that in the post. But to also clarify, I would like an answer in terms of the relevant mathematics. I think to me the two are quite intertwined. For instance, a satisfactory answer to the question of "what are spin-$j$ systems" would involve talking about how to classify irreducible projective representations of $SO(3)$. Assuming one knows introductory quantum mechanics, the only physics there is defining the correspondence between physical and mathematical objects. @MariusLadegårdMeyer $\endgroup$
    – Silly Goose
    Commented Jun 30 at 5:41
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    $\begingroup$ Agreed! I don't doubt a good answer will be mostly mathematical ;) $\endgroup$
    – Marius Ladegård Meyer
    Commented Jun 30 at 5:43
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    $\begingroup$ More on $SO(3)$ & $SU(2)$. $\endgroup$
    – Qmechanic ♦
    Commented Jun 30 at 6:33

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It's actually Spin(n) which is the universal and double cover of SO(n) for n>=3.

However, it turns out that Spin(3) is isomorphic to SU(2). This is important because SU(2) is defined via complex 2 × 2 matrices. Thus we can use complex matrices to represent elements of Spin(3). This is how the Pauli matrices come into the picture.

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