I haven't used Mendeley since I started using Papers, about 3 years ago. Mendeley at the time seemed to have an uncertain future and less than stellar performance. I find the Papers experience overall much better designed, though the 3.0 release, status post Springer's purchase of Mekentosj, still has some warts. Specifically, I found the PDF interaction is much better and with 3.0 the sync and mobile support is great.
That is a very relevant question, since Mendeley is free.
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.
The cost is incidental for many users, considering what is at stake -- basically, continued professional competence due to keeping up to date with the state of research.
How does Mendeley help to keep up with the state of research? I thought it just lets you manage the papers you read. Does it actually give you access to paywalled papers?
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?
I'm saying that the $50 or whatever that Papers.app costs versus the $0 Mendeley costs is not a factor for many rational actors (if those exist ;-). The real cost is the time and focus to find and read the articles. This swamps the $50. The $50 should not be a factor.
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.
Aim for journals with a high-impact factor. Researchers are encouraged to publish there. Unfortunately there's usually a pay wall for those. For free stuff you can check open-access journals. Some open journals also have a high impact factor and there's recently been a push for publish to publish only in open access journals.