The names are applied largely based on the structural characteristics of the dataset. To quote from @JeromyAnglim's answer there:
If we think of designs made up of $n$ cases measured on $k$ occasions, then the following loose definition seems to me to be descriptive of the distinction:
- longitudinal designs: high $n$, low $k$
- time series: low $n$, high $k$
To those two possibilities, we can add cross-sectional datasets. They have only one measurement occasion per case (i.e., $k=1$, but $N>1$ and hopefully reasonably large).
However, we certainly need to take those features into account when analyzing the data, no matter what we call it. The key aspect from a statistical perspective is that longitudinal / time-series data are not independent, whereas cross-sectional data may be. If existing non-independence isn't taken into account, any inferences will be invalid (e.g., confidence intervals may be too narrow).
- If you have cross-sectional data, standard, statistics-101-level analyses (e.g., a $t$-test), can be applied.
- If you have time-series data ($n=1$, large $k$), analysts will often use ARIMA methods to identify auto-regressive, integrative, and moving average terms.
- With longitudinal data ($k>1$ , but small), there usually isn't enough information to estimate ARIMA terms. Instead, people will typically use mixed effects models.