Easy differences arise if one allows principal ultrafilters, since the ultrapower of $X$ by a principal filter is canonically isomorphic to $X$, but other ultrapowers are not. Another easy difference arises when $I$ is uncountable, since one filter might concentrate on a countable subset of $I$ and others might not, and this can dramatically affect the size of the reduced power, making them different.
So the question is more interesting when one considers only non-principal filters and also only uniform filters, meaning that every small subset of $I$ is measure $0$.
In this case, under the Generalized Continuum Hypothesis, the ultrapower of any first order structure is saturated, and thus any two of them will be canonically isomorphic by a back-and-forth argument. Without the GCH, it is consistent with ZFC to have ultrafilters on the same set leading to nonisomorphic ultrapowers.
Also relevant is the Keisler-Shelah theorem, which asserts that two first order structures---such as two graphs, groups or rings---are elementarily equivalent (have all the same first order truths) if and only if they have an isomorphic ultrapowers.