An omnidirectional microphone has sensitivity at all angles – in principle picking up sound equally from all directions. This gives you extremely natural and realistic sound compared to a cardioid mic. A cardioid or any other directional microphone is much more sensitive to wind and pop (the plosives, i.e. the heavy air movement that comes with p-words) and thus requires a pop filter, which creates a larger footprint. Furthermore, a cardioid mic is also more sensitive to handling noise and placement due to the proximity effect. 

Nevertheless, a cardioid is typically the first mic that most people think of for a live performance with high amplifications and thereby a risk of feedback. DPA offers both omni and cardiod headsets. Both types can be used on stages with loud SPLs without having feedback issues. Not every location can use an omni due to the acoustical environment or loudspeaker placement, but having a linear and well-balanced PA system gives you the best possibility to offer very natural results from the omni with the smallest footprint on the face of the performer. DPA omnis are so linear and well controlled that risk of feedback is very low. As an example, we have seen actors in musicals performing elevated above the scene right in front of the PA speakers while singing their lungs out – without feedback!