When members of the same species are geographically isolated over time they will start to differ because of multiple factors:
1. Climate
2. Relief
3. Vegetation
4. Competition
5. Size of the area (specifically referring to smaller islands)
All of this factors (if different) will require a change in the behavior of the species, change in color, change in size, adaptation towards the food source by changing the teeth/beak and thus the jaw structure (for the species that have them), develop a new strategy to hunt (if it's a hunter), or develop a new strategy for evading predators, maybe change the niche in the food chain if there's stronger competition.