The formal Laurent series in one variables is
$$\begin{equation} \mathbb{C}((t))=\mathbb{Q}(\mathbb{C}[[t]])=\mathbb{C}[[t]][\frac{1}{t}] \end{equation}$$
What is the difference between this field and
$$\begin{equation} \bigg\{\sum_{n\in\mathbb{Z}}a_{n}t^{n}\quad\text{such that}\quad \forall n \in \mathbb{Z} \ a_{n}\in\mathbb{C}\}\bigg\}\end{equation}$$
In more variables what is the difference with this case in the definition?
The ring of formal Laurent series comprises formal series $\sum_{n\in \Bbb{Z}}a_nt^n$, where, for some $i \in \Bbb{Z}$, the $a_n$ with $n < i$ are all zero: so we could write the formal series as $\sum_{n=i}^\infty a_nt^n$. This reflects the fact that there is only one pair of square brackets round $\frac{1}{t}$ in $\Bbb{C}[[t]][\frac{1}{t}]$ - a notation that really means $\Bbb{C}[[t]][u]/I$ where $I$ is the ideal $(tu - 1)$. As mentioned in a comment this is the same as the field of fractions of the ring of formal power series $\Bbb{C}[[t]]$.
The element $tu - 1$ is a unit in the ring of formal power series $\Bbb{C}[[t, u]]$, implying that the quotient ring $\Bbb{C}[[t, u]]/(tu - 1) = \{0\}$. This reflects the fact that the convolution product doesn't make sense if you allow non-zero coefficients $a_n$ for negative values of $n$ of arbitrarily large absolute value.
If you are looking at formal Laurent series in more than one variable, the same considerations apply: e.g., $\Bbb{C}[[t, v]][\frac{1}{t}, \frac{1}{v}]$ means $\Bbb{C}[[t, v]][u, w]/(tu - 1, vw - 1)$ and comprises formal series $\sum_{m,n \in \Bbb{Z}}a_{mn}t^mv^n$, where at most finitely many $a_{mn}$ with $m < 0$ or $n < 0$ are non-zero.