How does quantile regression compare to logistic regression with the variable split at the quantile? I googled a bit but didn't find anything on this.
Suppose you do a quantile regression on the qth quantile of the dependent variable. 
Then you split the DV at the qth quantile and label the result 0 and 1. Then you do logistic regression on the categorized DV.
I'm looking for any Monte-Carlo studies of this or reasons to prefer one over the other etc. 
 A: For simplicity, assume you have a continuous dependent variable Y and a continuous predictor variable X. 
Logistic Regression 
If I understand your post correctly, your logistic regression will categorize Y into 0 and 1 based on the quantile of the (unconditional) distribution of Y.  Specifically, the q-th quantile of the distribution of observed Y values will be computed and Ycat will be defined as 0 if Y is strictly less than this quantile and 1 if Y is greater than or equal to this quantile.  
If the above captures your intent, then the logistic regression will model the odds of Y exceeding or being equal to the (observed) q-th quantile of the (unconditional) Y distribution as a function of X.
Quantile Regression
On the other hand, if you are performing a quantile regression of Y on X, you are focusing on modelling how the q-th quantile of the conditional distribution of Y given X changes as a function of X.  
Logistic Regression versus Quantile Regression
It seems to me that these two procedures have totally different aims, since the first procedure (i.e., logistic regression) focuses on the q-th quantile of the unconditional distribution of Y, whereas the second procedure (i.e., quantile regression) focuses on the the q-th quantile of the conditional distribution of Y.
The unconditional distribution of Y is the 
distribution of Y values (hence it ignores any 
information about the X values). 

The conditional distribution of Y given X is the 
distribution of those Y values for which the values 
of X are the same.  

Illustrative Example
For illustration purposes, let's say Y = cholesterol and X = body weight.  
Then logistic regression is modelling the odds of having a 'high' cholesterol value (i.e., greater than or equal to the q-th quantile of the observed cholesterol values) as a function of body weight, where the definition of 'high' has no relation to body weight. In other words, the marker for what constitutes a 'high' cholesterol value is independent of body weight. What changes with body weight in this model is the odds that a cholesterol value would exceed this marker.      
On the other hand, quantile regression is looking at how the 'marker' cholesterol values for which q% of the subjects with the same body weight in the underlying population have a higher cholesterol value vary as a function of body weight. You can think of these cholesterol values as markers for identifying what cholesterol values are 'high' - but in this case, each marker depends on the corresponding body weight; furthermore, the markers are assumed to change in a predictable fashion as the value of X changes (e.g., the markers tend to increase as X increases).         
A: They won't be equal, and the reason is simple.
With quantile regression you want to model the quantile conditional of the independent variables. Your approach with logistic regression fits the marginal quantile.
A: One asks "what is the effect on the nth quantile of the dependent variable's distribution?" The other one asks "what is the effect on the probability that the dependent variable falls into the nth quantile of its unconditional distribution?"
I.e., the fact that they both have the word "quantile" in them let's them look more similar than they are.
I guess if you first estimate a conditional quantile function, use this for the split and proceed from there, the two approaches would become more similar. But I don't see what you would stand to gain from such a detour.
.
A: This is roughly the deal if I've transcribed these correctly.  See https://en.wikipedia.org/wiki/Quantile_regression for $\rho_p$.
Logistic Regression:
$$
p(y_{thresh}) = \arg \min_{p} \sum_i J^{logistic}(p, y_i < y_{thresh})
$$
Quantile Regression
$$
y(p_{thresh}) = \arg \min_{y} \sum_i \rho_p(y_i - y)
$$
Question is (I can't remember) are the score functions for these variational problems the only ones possible for MLE? If not, is there a pairing that guarantees equivalence in the sense the the same pairings $(p, y)$ are generated?
