A complexity class separation proof uses uniformity of complexity classes essentially if the proof does not prove the result for nonuniform version, for example proofs based on diagonalization (like time and space hierarchy theorems) make essential use of uniformity as they need to simulate the programs in the smaller class. Which results in complexity theory (other than diagonalization proofs) use uniformity essentially?
Which results in complexity theory make essential use of uniformity?
cc.complexity theory
We suspect Permanent requires superpolynomial-size circuits (in either of the arithmetic or Boolean models).  However, if we consider Boolean circuits with threshold gates, currently we can only prove superpoly lower bounds in the case of depth-restricted, uniform circuits.  I believe the most recent reference for results of this type isA Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanentby Koiran and Perifel.(Their proof involves diagonalization at some point, so this doesn't strictly speaking meet your criterion, but I thought it might still be of interest.)