Welfare cost of business cycles

The difference between conditional and unconditional welfare is not about the mean, it is about the conditioning set for the expectations operator. The mean aims at computing an expected value. That expected value can either be the unconditional expected value

or a conditional expected value, i.e. conditional on the information at time t.

The unconditional mean (as e.g. saved in oo_.mean) can be interpreted as integrating out initial conditions and future shocks. In contrast, the conditional mean will take the starting point in the state space at time t into account and only integrate over future shocks.
For that reason, the conditional welfare takes into account e.g. the amount of capital you have at time t when you want to compute conditional welfare. Taking the steady state as the conditioning set is a common choice, but only one of infinitely many potential choices at which to evaluate conditional welfare.