Is there any advantage to use $\bmod q$ over naïve method (without using modulo)? If yes, is it security or computational complexity or any other?
Yes; doing things $\bmod q$ does have the practical advantage that the shares are bounded length; computing the shares in $\mathbb{Z}$ can potentially have us send rather long values (as the values there don't have an upper bound).
However there are also security concerns:
Revealing the shares in $\mathbb{Z}$ leaks information; for example, suppose someone knows the share $(x, y)$ for $x=2$ that corresponds to a secret $z$. That is, he is given a value $y = a_n2^n + a_{n-1} 2^{n-1} + ... + a_12^1 + z$. Now, the nonconstant terms are all even; hence if they see that $y$ is odd, that means that $z$ must also be odd; that is, we just leaked the lsbit. Extending this observation shows that a share $(x, y)$ reveals the value of $z \bmod x$. A similar observation shows that two shares $(x_0, y_0)$, $(x_1, y_1)$ also reveals $z \bmod x_0 - x_1$. This is in contrast to the standard Shamir Secret Sharing, which has no such leakage.
Shamir assumes that the secret coefficients $a_n, a_{n-1}, ..., a_1$ are chosen uniformly. However, it turns out to be impossible to select uniformly randomly from a set of size $\aleph_0$ (which the set of integers is), that is, any selection method must necessarily be biased. And, depending on what the distribution is, this bias will also leak further information.
BTW: Shamir Secret Sharing isn't necessarily done modulo a large prime; it can be implemented over any finite field. In practice, we often use even characteristic fields, such as $GF(2^8)$ or $GF(2^{128})$; the security is the same, but it has the practical advantage of everything fitting in an integral number of bytes.
