The choice between "traditional NIS" or the NIS code in the NYS library is a choice between laziness and maturity vs. flexibility and love of adventure.
The "traditional NIS" code is in the standard C library and has been around longer and sometimes suffers from its age and slight inflexibility.
The NIS code in the NYS library requires you to recompile the libc library to include the NYS code into it (or maybe you can get a precompiled version of libc from someone who has already done it).
Another difference is that the traditional NIS code has some support for NIS Netgroups, which the NYS code doesn't. On the other hand the NYS code allows you to handle Shadow Passwords in a transparent way. The "traditonal NIS" code doesn't support Shadow Passwords over NIS.
Forgot all this if you use the new GNU C Library 2.x (aka libc6). It has real NSS (name switch service) support, which makes it very flexible, and contains support for the following NIS/NIS+ maps: aliases, ethers, group, hosts, netgroups, networks, protocols, publickey, passwd, rpc, services and shadow. The GNU C Library has no problems with shadow passwords over NIS.
The choice between NIS and NIS+ is easy - use NIS+ only if you have severe security needs. NIS+ is much more problematic to administer (it's pretty easy to handle on the client side, but the server side is horrible). Another problem is that the support for NIS+ under Linux contains a lot of bugs and that the development has stopped.