A stochastic process is a series generating mechanism.
A time series is a particular realization of a stochastic process.
A stochastic process is a random mechanism that generates the time series data and the realization of a stochastic process is a specific observed path.
An economic time series can be thought of as a specific realization of an underlying stochastic process.
It is important to emphasize that the key implication of assuming that the data is generated by a stochastic process is that we allow for dependence in the data across different periods (instead of assuming the data as iid)