What does papers provide over Mendeley?
Reading the Web site the only advantage that I found is that it allows you to use your Dropbox account to sync the files, while for Mendeley, if you want more space than a free account offers, you have to pay.
By the way, as a non-academic, I've always wondered how I could keep up with the latest developments in the field of information security/cryptography. Are there simply some key journals that one has to keep track of, or are there other more frequently updated sources that one should read?
If one or the other app happens to have features, tradition, or lock-in, in its favor, so that it will better turn your time and focus into increased subject mastery, that's what really matters.
I've found that having my papers organized by subject matter, searchable, and always available on any device, is a huge advantage in learning new stuff.
Over the years, it adds up, and you end up knowing more, which is helpful in lots of ways. To be crass -- it could be the difference between a 4% raise and a 5% raise for someone making $100K/y ($1K in the first year). That's a big deal.