In the second spectral sequence you write $R^q f_\ast I^\bullet$. This is correct but $R^qf_\ast$ must be considered as the $q$th hyper-derived functor applied to the complex $I^\bullet$, it is not the ordinary derived functor $R^qf_\ast$ applied termwise. See my answer to Construction of the spectral sequence of Katz/Oda for more details.
Once you observe this, then you find that the hypercohomology spectral sequence is just the special case of the Leray spectral sequence when $f = \mathrm{id}$. Indeed the hyper-derived functor $R^q \mathrm{id}_\ast$ is just the functor $\mathcal H^q$.