We have both.

Some of the advantages of hard side (of course it depends on the piece)

It's waterproof
it can be lighter
it's significantly easier to clean
you can stack another bag on it
it is sturdier


and soft side

expands when you need more capacity
collapses for storage
usually has outside pockets
Can have several compartments
Doesn't usually show wear as easily


We also have a hybrid bag (or did, not sure if it's in the current rotation or if it died) that has a hard bottom and sides, but a soft top to allow for the benefits of both.