What's the benefit of Upcounting or Downcounting or Center-aligned mode of timer/counter?
It depends on your application!  Up counting is the most common when counting intervals.  Down-counting is much more practical when performing count downs before an event occurs, because you start from where your want down to zero instead of offsetting your trigger relative to your current count or both reset timer and trigger to desired interval.

I personally never used the center aligned mode, but based on a STM App note (DM00042534), it can be used to generate asymmetrical PWMs with a given phase shift...