I don't see the difference in the 'statistical' and 'empirical' approaches.
Statistical or Data Mining or Machine Learning approaches, which mostly are under the same umbrella, rely on inductive inference. On the other hand, the analytic approach, relies on some prior axioms which we assume to be true by definition, and beyond this step, the theory is constructed in a deductive manner - i.e. as the implications of the particular set of axioms about human behavior. Equilibrium approaches derive from economic theory, or financial economics. CAPM is linked to assumptions about utility, hetero/homogeneity of agents and their expectations. To what degree any of these assumptions are 'true' (or not), is something the testing of the theories has to answer (and to a certain extent what would be 'reasonable' to expect of human behavior).
The 'law of one price' is something which we can subscribe to more easily than say the alphas of a CAPM regression of stock returns on market returns, being jointly zero (in a statistical sense), the latter being a particular case of consumption covariance risk, and all consumption risk being proxied for by the market.
In this sense, the applicability of analytic approaches is limited to operating within the confines of the axioms which lie beneath the theory - or expecting violations of those axioms in particular cases (which perhaps over time will expand that theory to a newer one).
However, as more complex/realistic theories are devised, there is also the concern whether the theory itself was formed after peeking at the data - i.e. devising theories to explain persistent patterns or anomalies which an earlier theory could not 'explain' away. In this context, Fama-French's model is not a theory - it spotted an empirical regularity which was not explained by CAPM, but it is not a theory in the deductive sense. However, CAPM is a theory - albeit one which does not explain (far less predict) accurately all aspects of real markets.
The application of analytic approaches to derivatives pricing is readily understood because the law of no arbitrage is easier to swallow as a 'law' than say equilibrium arguments about the market. The 'purest' analytic approach is the one which tells me to buy oranges in market A at price 'p1' and sell them in market B at price 'p2', where p2 > p1. Short of this, we are always estimating some parameters even to use what is otherwise a deductive theory.
Also, I suspect that the statistical and analytic approaches are not always mutually exclusive. As an example, if we accept CAPM to hold more or less, the series constructed from the residuals could be then subjected to torture by machine learning methods to extract a signal, if any. In this case, the lines are blurred.
If we think of this as a Bayesian, then the 'analytic' or theoretical arguments help us form our prior, and the data we have at hand is 'blended' with our prior to arrive at the posterior.
At one extreme, 'pure' statistical/data mining approaches are left to function without any prior structure we impose on the market dynamics and solely on the detection of empirical regularities (though, even using a trader's inputs to a data mining approach would constitute becoming somewhat 'analytic' - as we expect the trader to have some sort of prior story to justify his inputs). At the other extreme, pure analytic example would be exploitation of LOP as in the oranges example - i.e. the inference holds by force of logic.
But the question of which approach is dominant - I suspect, as Dirk pointed, these things ebb and flood, and moreover, 'reasonable' assumptions giving some 'weak' structure to a 'data mining approach' is probably more profitable when we can posit some structure, instead of no structure at all.
p.s: nice answers by QuantGuy & Dirk! as expected.