Downspouts that are buried by way of a rubber boot is a common method of directing the rain (or melted snow) away from the foundation of the building and typically under a walkway so as not to put water where it can wet or refreeze and become a hazzard.
It is important to extend the underground spout to a teminus at ground level for proper discharge. This can be among bushes or in gravel or rock areas. You do not simply dig a hole and let water accumulate. It won't work and will ultimately erode that ground around it.
Be certain to keep the boot and extension clean and able to handle the water.
Another of the advantages is that the asethetics are much neater and cleaner.
A disadvantage is if the boot or extension becomes clogged or damaged, it can be more involved to repair.