A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?
Because many troubling discrepancies are showing up in comments and answers, let's refer to some authorities.
James Hamilton does not even define a time series, but he is clear about what one is:

... this set of $T$ numbers is only one possible outcome of the underlying stochastic process that generated the data.  Indeed, even if we were to imagine having observed the process for an infinite period of time, arriving at the sequence $$\{y_t\}_{t=\infty}^\infty = \{\ldots, y_{-1}, y_0, y_1, y_2, \ldots, y_T, y_{T+1}, y_{T+2}, \ldots, \},$$ the infinite sequence  $\{y_t\}_{t=\infty}^\infty$ would still be viewed as a single realization from a time series process. ...
Imagine a battery of $I$ ... computers generating sequences $\{y_t^{(1)}\}_{t=-\infty}^{\infty},$ $\{y_t^{(2)}\}_{t=-\infty}^{\infty}, \ldots,$ $ \{y_t^{(I)}\}_{t=-\infty}^{\infty}$, and consider selecting the observation associated with date $t$ from each sequence: $$\{y_t^{(1)}, y_t^{(2)}, \ldots, y_t^{(I)}\}.$$  This would be described as a sample of $I$ realizations of the random variable $Y_t$. ...

(Time Series Analysis, Chapter 3.)
Thus, a "time series process" is a set of random variables $\{Y_t\}$ indexed by integers $t$.
In Stochastic Differential Equations, Bernt Øksendal provides a standard mathematical definition of a general stochastic process:

Definition 2.1.4. A stochastic process is a parametrized collection of random variables $$\{X_t\}_{t\in T}$$ defined on a probability space $(\Omega, \mathcal{F}, \mathcal{P})$ and assuming values in $\mathbb{R}^n$.
The parameter space $T$ is usually (as in this book) the halfline $[0,\infty)$, but it may also be an interval $[a,b]$, the non-negative integers, and even subsets of $\mathbb{R}^n$ for $n\ge 1$.

Putting the two together, we see that a time series process is a stochastic process indexed by integers.
Some people use "time series" to refer to a realization of a time series process (as in the Wikipedia article).  We can see in Hamilton's language a reasonable effort to distinguish the process from the realization by his use of "time series process," so that he can use "time series" to refer to realizations (or even data).
