There is an overlap, in that both words can describe an emotion, especially a more elevated or delicate one like love, awe or compassion. But if you are talking of a physical sensation, like pain or hunger, then it has to be "feeling". And "feeling" would generally work better for the rawer emotions like anger or lust. A sentiment can also be more the result of a feeling than the feeling itself. It may refer to more of a mindset, opinion or attitude. People talk for example of "public sentiment" in response to some event, such as an economic crisis or terrorist attack.

Eu acho que "sentiment" é mais general, mais etereo. "Feeling" se foca mais no como a pessoa, well, "feels" (sentir-se?). "Sentiment" pode focar-se no motivação o intenção, "feeling" se foca mais no resultado.