The Photos app on your iPhone, iPod touch, iPad and Mac lets you use the Auto Enhance and Remove Red-Eye tools with Live Photos in a non-destructive manner, which is nice.
But use filters, or perform a crop or trim on your Live Photo and it'll get turned into a still image because Apple's Photos solution isn't advanced enough (yet) to support these editing operations in conjunction with Live Photos.
This can be quite cumbersome for two reasons. A) You must manually restore your Live Photo by tapping Edit → Revert; and B) these stills only include the motionless part of your Live Photos.
With Live Crop, a nifty little app by developer MobiLab, not only can you trim and crop out parts of your Live Photo without destroying its live component, but also crop, zoom, trim and resize Live Photos, your videos as well as animated GIFs.