You are also potentially giving up flexibility, depending on what you need to run. A router is pretty much only going to work for backups (and generally only if you have the backup software running on the client, or are doing it manually), file sharing and video/audio streaming to an appropriate client.
Even a cheap NAS is likely to do that better than any router, and likely to have additional capabilities (yes, I know a very tiny number of routers also have things like iTunes servers, torrent servers and download servers. A TINY number and those routers are generally more expensive than most low end NAS).
A high end, but still very consumer NAS, is likely to have very good performance and very flexible in terms of what it can run.
Still pales compared to a full server running a "full" OS like Linux or Windows in terms of flexibility and performance, but for a lot of uses cases it can be as good or better (better could mean support, ease of deployment, lower power budget, etc.). I personally go the full, but very low power, server route because I need the extra flexibility as well as wanting the extra performance (I don't see a NAS that can do SMB3+SMB Multichannel yet, and probably never will and a 10GbE NAS is WAYYYYY more than my server).
At some point I'll probably get a cheap NAS to run as a back-up machine, because it is cheaper and less expensive in the long run to operate than building another server, and even when I "retire" most of my server components to build a newer box, it is probably more cost effective to sell off the old parts and get that cheap NAS than it is to continue to utilize them as a backup machine (though debatable, as it'll probably be in S5 most of the time, except for a few minutes every day or two to be powered up, take a backup and then power back down. I love WOL from S5 :-D).