Specifically, it was recently shown that the fastest implementations of ORAM (https://en.wikipedia.org/wiki/Oblivious_RAM) are actually in line with the lower bounds of theoretical Differentially Private RAM performance - meaning you gain zero performance advantage by using a weaker security model. Of course, it's unclear if this is true for other data structures and usages, but ORAM is a non-insignificant research area.
See https://eprint.iacr.org/2018/1051 for more details.