I would choose Affymetrix because of the huge number of open-source tools available for various analysis. Illumina DNA arrays never achieved that kind of popularity. They might still work though. You will also have to consider cost.

You might want to check this review and some of its back-references to know how the high-throughput platforms are compared:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3605921/

It depends on the species you analyze. If humans, probably what Sameet Mehta answered is true. If animals, then most techniques are on the Illumina side.. Also density (and cost, of course) could be a good parameter to choose.