I asked my professor, and the answer appears to be differences in both the generation and the final product.
Free chromosome fragments are created through irradiation/other damage of the germline in one animal. Through a series of crosses, it is possible to introduce individual fragments (containing a duplication of your gene of interest, as well as a marker) into a mutant background. The extrachromosomal fragment will be lost during mitosis at various points in development, and in a mosaic analysis, you can look for the loss of your marker (and therefore the loss of the wildtype allele), and see if yfg product is required in certain cell types for a wildtype phenotype.
Extrachromosomal arrays are created by cloning, and you typically have many copies of both the gene and marker. It's the same principle as free chromosomal fragments in terms of mosaic analysis, but it's very powerful, as you can easily clone individual genes (versus irradiating and hoping to discover yfg on a fragment) and markers (and now you can use non-endogenous markers like GFP).
Even though extrachromosomal arrays are comparatively easier to generate, the problem is that you end up with many copies of the gene, which leads to non-endogenous dosage levels. Even if this doesn't cause a completely different mutant phenotype in the cells, it is possible that enough of these proteins could segregate at mitosis into the daughter cells, such that even if the fragment is lost, you could still have your gene product around in the daughter cells (if your marker protein didn't also segregate like this, you might draw erroneous conclusions about the necessity of yfg product in those particular cells).