Under similar conditions, is melting faster than solidification? Not the same condition, only similar condition.
I recall reading about a statement in a textbook (unfortunately, couldn't find the source now) saying the change from disorder state to ordered state is always faster than the other way around, an example is melting is faster than solidification.
So, let's suppose two identical copies of a system, both at identical conditions: a phase equilibrium at $(T, P)$ between solid and liquid (of a certain substance). 


*

*In copy A the environment temperature is increased to $T + \delta$ (so
melting occurs)

*In copy B the environment temperature is decreased
to $T - \delta$ (so solidification occurs).


Do both have the same rate of phase transition (measured by, say, moles per minute)? Thermodynamics does not tell rate of change.
 A: 
Thermodynamics does not tell rate of change.

That is correct. As it is often said in the context of (chemical) reaction kinetics: 'thermodynamics says nothing about kinetics (rate)'.
In the case of melting and solidification, the Standard change of Gibbs Free Energy $\Delta G$ is the same for both processes, that is:
$$\Delta G_{\mathrm{m}}=-\Delta G_{\mathrm{s}}$$
But this tells us nothing about the rate of melting and solidification.
This is evidenced empirically by a phenomenon called supercooling. If a pure substance is cooled down fairly slowly it's quite easy to achieve a temperature well below the melting point of the substance, without any solidification occurring. It's quite easy to achieve the supercooled state with distilled or deionised water, for instance. So during supercooling the rate of solidification is effectively zero.
There is no equivalent (as far as I know) of a solid substance not melting when it is heated above its melting point.
Intuitively this is not hard to understand: heating causes the molecules (or atoms or ions) that make up the crystalline lattice to increase their kinetic energy, until they are energetic enough to break away from the lattice.
But the reverse process requires these particles to find the right position to 'dock' onto the existing lattice.
A well known supercooled 'liquid' is glass: at room temperature it is a highly (almost infinitely) viscous liquid, without any crystalline structure (it's amorphous).
A: As thermodynamics cannot answer this question, we need to consider something else. 
First, the temperature changes in your post might not happen.  If the state is between the two saturated points at certain pressure level, adding or removing heat will not change the temperature but will change the mixture towards solidification or welting direction. 
The rate can be determined by many factors. So it has to be in context. Assuming heating from the boundary, the heat will transfer at certain speed. We know liquid water heat conductivity is about 0.6W/m-K and ice heat conductivity is about 1.6. So heat transfer is faster in ice so you can imagine that ice will melt faster. However, this is only one factor. There could be many other factors. 
