If you allow singularities, it is easy to construct such examples. For example, given any birational class of surfaces, you can find one with a rational double point of fixed type (say $A_n, D_n, E_6.E_7$ or $E_8$). But these have only one analytic type, so taking different birational classes, you can find such local rings which are not isomorphic, but their completions are. It is somewhat more subtle to do the same if you fix the birational class and in general it is impossible except in the case of rational surfaces, when in some cases you can find such non-isomorphic ones. For example, if $R$ is the local ring at the origin of 3-space, $R/(x^5+y^3+z^2)$ and $R/(x^4y+x^5+y^3+z^2)$ both are $E_8$ singularities on a rational surface, non-isomorphic, but of course their completions are isomorphic.

If you look at local rings, it is easy to construct such examples. For example, if $X,Y$ are smooth of same dimension, then for any point $x\in X,y\in Y$, the completions of $O_{X,x}, O_{Y,y}$ are isomorphic, but of course the algebraic local rings are not necessarily isomorphic (for example, if $X,Y$ are not birational, then even their fraction fields are not isomorphic.)