The paper log is the only physical "evidence" that the cacher was indeed at the cache site, found the cache, and signed the log to note that find. If the online log was the only requirement, cachers could falsely claim a find by just submitting a "Found It". Not all online logs specifically state where they found the cache, what type of hide it was, what was in the cache, etc. If the online logs contained those items, they could also be used as further hints for future finders ("Spoilers" as is said in the business).
I'll leave you with one last comment -- ALWAYS BRING A PEN/PENCIL WITH YOU!! You'll never know when you'll need it.