What are the differences between GWAS and QTL mapping? What are the differences between the two methods? 

What advantages does one have over the other, and what are their limitations relative to one another?
The basic difference between GWAS and QTL mapping is that GWAS studies the association between alleles and and a binary trait, such as being a sufferer of a disease, while QTL analysis deals with the contribution of a locus to variation in continuous trait like height.

Thus GWAS normally deals with frequencies (i.e. is the frequency disease probands greater in those with the minor allele compared to the major allele at locus X), whereas QTLs deal in correlations (Is the number of minor alleles at locus Y correlated with height). 

Methodologically the primary difference would be the distribution that links the allele to the outcome (disese/continuous phenotype). GWAS is going to use Chi-squared statistics or logistic regression, QTL with more likely use straight up normal distribution based linear regression.