How do these differ? I've been pronouncing them exactly the same.
Oooh! Good question. I haven't thought about it in some time, but the Japanese sound is pronounced (at least in my experience in Tokyo) with the tip of the tongue closer to the top front teeth, as in the Mandarin xi sound. The way I pronounce English "she" as a native speaker is with air over the middle-front of my tongue, not the tip.
As it so happens, the Edokko accent pronounces hito like shito; this may offer you a way to approach the shi sound in standard Japanese (if you have hi covered). You can try aspirating hi more strongly until you get closer to a shi sound that passes muster.