Why would the Camargue horses have a white coat while the foal have a dark coat?
Our own hair will also turn gray (indeed, yours seems to be on its way, if you don't mind me saying so) but there need not be an advantage to that; if it doesn't affect our ability to have successful children it probably doesn't matter much to evolution or selection.  When it comes to animals that interact with humans, though, many traits are selected for, even those without an advantage, in fact often we select for disadvantageous traits (if they were advantageous, we might not need to select for them in the first place).

As this 2008 paper from Nature Genetics goes into, the "graying with age [phenotype] is an autosomal dominant trait" overwhelmingly caused by "a 4.6-kb duplication in intron 6 of STX17 (syntaxin-17) that constitutes a cis-acting regulatory mutation".  To answer your question, though:


  The Gray horse provides a notable example of how humans have cherry-picked mutations with favorable phenotypic effects in domestic animals... The prestige of riding a white horse has thus led to selection of the Gray-causing mutation by humans; this mutation is by far the most common cause of white color in horse.


This phenotype is actually often detrimental, as white/gray horses can have eye problems and higher rates of skin cancer; Camargue's are particularly susceptible to melanomas, present in around 70% of horses by age 15.  There does seem to be some benefit, though, as the second paper I linked suggests these horses are subject to fewer fly attacks.

EDIT: I seem to have been unclear and you seem to have misunderstood my answer.  Let me break it down:


Camargue horses are gray.
Gray color is a simple mendelian autosomal dominant phenotype caused by a loss of pigment in adulthood.
Gray color is generally bad but sometimes good; either way, it's probably not a huge detriment.
Camargue horses are homozygous for gray color.
Humans have in the past selected for gray color in horses in general.


That is an example of why something negative or pointless might persist, as would a dominant inheritance pattern that is fixed in a population: negative traits that are only weakly deleterious have a non-negligible chance of fixation.  Asking what the advantage is is somewhat of a moot question because every single Camargue horse has the trait; asking what caused a seemingly-negative phenotype to become fixed is a good question, but likely an unanswerable one as it occurred long ago.

As a side note, how long a breed has been in an area does not simply imply breed purity.  Here's a quote from the International Museum of the Horse:


  Through the centuries many armies have passed by the Camargue, including the Greeks, Romans and Arabs. The horses brought with these armies influenced the Camargue over time. It has even been suggested that the Camargue has had some influence on the early breeds in Spain as armies took them back home.