By [Pólya’s theorem][1], any even real-valued function $f$ on $\mathbb R$ with $f(\infty-)=0$ which is convex on $[0,\infty)$ is positive definite. So, any such function is the (auto)covariance function of a stationary Gaussian process; see e.g. [Section "Properties of the Autocovariance Function", page 2][2]. 

Now just take any two different functions, $f_1$ and $f_2$, of the Pólya class such that $f_2(t)=1-|t|=f_2(t)$ for $|t|\le1/2$. Then the corresponding stationary Gaussian processes, say $(X_{1,t})$ and $(X_{2,t})$, with the covariance functions $f_1$ and $f_2$ will have different distributions. Therefore, these two processes will be different from each other. 

  [1]: https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
  [2]: http://www.stat.columbia.edu/~rdavis/papers/VAG002.pdf