What varied between the finches of each island?

Through natural selection. Certain beak types are favoured for certain food - large short beaks are better suited to eating tough seeds, while long slender beaks are better off for eating insects and small seeds. Short answer -> finches adapt by expressing different allele frequencies when exposed to different environments. The differing allele frequencies translate to different beak phenotypes, which can be advantageous depending on the environmental factors, such as the food available.