That depends mainly on three factors: interface, resolution and speed.
If you're using the L1020, for example, Microsoft Camera would launch a lot faster than Lumia Camera (~1 second vs ~3.5 seconds). Microsoft Camera is limited to taking 5 MP photos only, while Lumia Camera can, of course, exploit the high-resolution sensor. The time between shots is also considerably longer with Lumia Camera. As for newer phones, firmware updates were supposed to have improved the performance of the Lumia Camera (including launch time and shot-to-shot interval), but the Microsoft Camera is still faster than the Lumia Camera app. The user interface of the Lumia Camera is also much better, allowing the user to fine-tune his or her shot intuitively.
Interface: Lumia Camera wins
Resolution: Lumia Camera wins
Speed: Microsoft Camera wins
So, if you're taking a quick snap of something fleeting - use Microsoft Camera. If you're composing an artistic high-quality photo, use Lumia Camera.
I use Lumia Camera most of the time.
P.S. Microsoft Camera is no longer available in newer devices... so that should give us a hint.