What is the difference between the standard representants of $\mathbb Z/q\mathbb Z$?
When decrypting in lattice-based cryptosystems, one computes a value $v \in \mathbb{Z}_q$ that is guaranteed to be congruent to a "small" integer $e \in \mathbb{Z}$, where $e$ encodes the message (e.g., as the parity of $e$ modulo 2).  By using the integer representatives between $-q/2$ and $q/2$, one can recover the small integer $e$ (and thereby recover the message) simply by "lifting" $v$ to its integer representative.

If we instead used the representatives $0$ through $q-1$, then the representative of $v$ would often be a little smaller than $q$ (specifically, when $e$ is negative), and would not have the proper parity.

Geometrically, using the correct representatives corresponds to "decoding" the lattice $q\mathbb{Z}$ under small error: given the coset $(v + q\mathbb{Z}) \in \mathbb{Z}_q$, we find the smallest element in the coset, which is $e$.

Many papers conflate $\mathbb{Z}_q$ with its set of representatives, but this can cause confusion about what operations are legal and meaningful. Values that are known to be "small" should be seen as elements of $\mathbb{Z}$, which are then reduced mod $q$ when combined with values from $\mathbb{Z}_q$. When we later want to recover a small value from its coset modulo $q$, we lift/decode to $\mathbb{Z}$ using the proper representatives.