"To sense" is to detect with one of the senses (sight, hearing, touch, smell, taste). When one touches something she feels it. So "feel" and "sense" are related. Here are some examples. I feel the cold (temperature). I sense that you are angry. = I feel that you are angry. [Both meaning "I can see that you are angry."] I sense danger in this location, and I feel afraid. I feel like eating dinner now. [INCORRECT: I sense eating dinner now.]

In some meanings, these are the same thing. In others, not so much. "I get the sense that you don't want to be here." "I get the feeling that you don't want to be here." In this instance, they're the same thing. In general, when you're talking in a figurative sense, they're more or less synonymous. "Reach in there. Do you feel it?" In this case (a strictly physical sense), "sense" can't be substituted here. So, "feel" can be used for the physical and non-physical, while "sense" can normally only be used for the non-physical.