I know that an $I(1)$ process becomes stationary after differencing once. However, I somehow always equated that to its being a random walk because say having a unit root process like \begin{eqnarray} y_{t} = y_{t-1} + \epsilon_t \end{eqnarray} is clearly $I(1)$ and becomes converted to an $MA_\infty$ $y_t = \sum_{s=0}^t \epsilon_{t-s}$, which strikes me as a random walk as discussed here for example.
Now, is there a difference between a random walk a process $I(1)$?