I would like to add to JamalS's answer that the way I see this is that the energy operator is the generator of time evolution. A generator, in this context, means that we exponentiate it to get the transformation, $\hat{U}=e^{-\frac{i}{\hbar} \hat{E} t}$.
The Schrodinger equation becomes an operator equation between the time-evolution generator and generators of other transformations, so that $\hat{E}=\hat{H}$ is just writing $\hat{E}$ in terms of other operators (classically, we also use $H$ to write the energy in terms of $p$ and $x$).
So, if we say that $\hat{H}=\hat{p}$ for example, we then plug it into Schrodinger's equation to find that the generator of time evolution is $\hat{p}$, which is the generator of translations in space, so as time evolves everything would just shift with the same speed of 1. The usual $\hat{p}^{2}$ will just case dispersion. Having an $\hat{x}$ means that now momenta will shift because $\hat{x}$ is the generator of translations in $p$ just like $\hat{p}$ is for $x$.
Another interesting example is when we have an spin in a magnetic field $\hat{H}=-m \hat{\vec{\sigma}} \cdot \vec{B}$, where $\hat{\vec{\sigma}}$ are the Pauli spin matrices, and $m$ is the magnetic moment. Here then well have $\hat{E}$ being equal to something proportional to $\hat{\vec{\sigma}}$, which is the generator of spin rotations. We can immediately predict that in a magnetic moment an spin will rotate (precess) in some uniform fashion!
By the way, this only works if $\hat{E}$ is explicitly independent of time. If not, one cannot directly make the connection that it is the generator of time evolution and one has to generalize this (involves using Dyson formula and other things).