There are two main advantages:
- Data safety (a NAS is especially useful if you have one with 2 or more bays. Like that you can put them in a raid easily and let them mirror the data. So if one of the drives fails, you can just replace it and the NAS will re-mirror it from the other drive automatically (or depending on the RAID version you are using different other things)
- access over the network with different devices. This means having your data available on your laptop, you desktop and the tablet of your wife at one central place. This is even possible over the internet, if you are willing to make it available via internet (but you do not have too. It is of cause more secure to just keep it in the LAN)
- pretty good automatic backup solutions are available, the NAS will keep track of your disc health and tell you if you need to think about changing the drives, etc.
- you can handle access rights very easily... so for instance one user is allowed to open one folder, the other one isn't and so on.
The result is:
- your computer can die and you still will have the files available.
- (at least) one of the drives can die if you have a 2- or more bay-system
- think of rendering with A:M... just put your project on the NAS and open A:M on each of your computers to render from it.
- it is a little less fast than a drive in your computer (at least as a SSD if you are "only" using 1 Gigabit-LAN / ethernet... if you have more like 10 or 50 and a fast raid configuration it will not make a big difference anymore). Anyhow it is still pretty fast and for most things you won't notice a difference. 4k video editing or something like that might be one of the things you might not want to do with a normal NAS, but other stuff isn't going to be noticeable slower.
- you need two drives (or for instance 3) and will loose 1/2 or about 1/3 (raid 0/1 vs. raid 5) of the capacity of your drives.
I am using a 4-Bay Qnap-NAS for all my data here for years now and I really love it and just recently have gone from about 5 TB of drive space to about 9TB of drive space because of video footage I am done with which is saved there. While working I might copy it to one of my SSDs and later the stuff goes to my NAS. 1080p material can even be edited on my NAS pretty nicely.