In Equity or FX, you are modelling the dynamic of a single number $S_t$ the stock price or fx spot rate.
In interest rate, you are modelling the dynamic of a curve $(f(t,t+\theta))$ all the forward rates (equivalently all the discount factors) for all tenors $\theta$.
So mean reversion, in the context of rates should be mean reversion around a mean curve. This mean curve is the forward curve built from the prices of deposits, futures and swaps observed in the market.
In the Vasiceck model the short rate mean reverts to a long term mean $r_\infty$. This means that you are mean reverting to a flat curve. But curves observed in the market are never flat and the model cannot even fit the forward curve observed at time 0. The Hull-White model improves on this by allowing you to fit the current expectation of the rates curve and mean revert to it. This requires replacing two of your parameters (the initial and long term short rates) by a whole curve (the initial forward curve which appears in the model in the form of the tenor dependent mean $\varphi(t)$).
Hope this helps