There is a connection! Keating and Snaith make their conjecture by modeling the distribution of $\zeta(s)$ by the distribution of the characteristic polynomial of a random unitary matrix, distributed according to Haar measure. This is connected to symmetric function theory by Schur-Weyl duality.
To my knowledge the first time this connection was used to compute moments (of a characteristic polynomial) was in a beautiful paper of Bump and Gamburd. See in particular Prop 4 (and perhaps especially the quick proof) and Corollary 1.
As to other connections, slightly different averages of the zeta function, such as $\frac{\zeta(\alpha+s)\zeta(\beta+\overline{s})}{\zeta(\gamma+s)\zeta(\delta+\overline{s})}$ are also be conjectured to be related to analogous averages in random matrix theory, the computation of which can be reduced to symmetric function theory in a similar way. This too is done in later sections of the Bump and Gamburd paper.
There is also a very nice paper of Dehaye that you might find interesting expressing lower order terms in conjectures for the moments of $\zeta(s)$ in terms of counts of taubleaux.