In many cases they could be interchangeable. But, in my opinion:
Research is more like finding out information. You could research a historical event, what restaurants are the best in a certain area, what company you want to work at, etc.
Study is more like memorizing or becoming familiar with information. You might study, English, math, etc.
For example, you usually study a vocabulary word (how to spell it, it's definitions, etc.). If you research a word that's more like you want to know it's origins, etc.