By using a battery pack, you can ensure that your phone does not get bulky and you can use a higher capacity power bank to charge your phone many times. The problem though is that you will also have to carry the battery pack with you at all times along with a charging cable. The problem here is that battery packs are quite heavy. With a phone case, you don't need to carry separate wire and your phone can easily work 2-3 days depending on the capacity of the battery case. In addition to this, it also adds an extra layer of protection to your phone since the case is also big and thick. But this makes your phone bulky to carry around. And compared to a power bank it is much more expensive. So the answer is that it would depend upon your own requirements.