Such an example is given by the pair of fibrations
$$
K(\mathbb{Z}/2,1) \to K(\mathbb{Z},2) \to K(\mathbb{Z},2)
$$
(coming from the Bockstein exact sequence) and by
$$
K(\mathbb{Z}/2,1) \to K(\mathbb{Z}/2,1) \times K(\mathbb{Z},2) \to K(\mathbb{Z},2).
$$
(a trivial fibration).

In both cases, the (cohomological!) Leray-Serre spectral sequence is concentrated in even total degree.  However, the two spaces have differing $H^2$ with integral coefficients ($\mathbb{Z}$ versus $\mathbb{Z} \times \mathbb{Z}/2$).