NRLMSISE is an empirical model using solar activity observations (mainly the F10.7 solar flux, i.e. the flux at a wavelength of 10.7cm, but also other observed variables).
The Nasa Standard atmosphere is a static model. It is dependent on time and location as well but not on actual observations of the solar activity, geomagnetic activity, etc.
This means that NRLMSISE-00 will be more accurate in general (although still often wrong by about 25%).
Aside from this, NRLMSISE-00 also outputs the density of several components of the atmosphere, such as Helium, O2, N2, etc.
Finally, the Standard Atmosphere describes the whole atmosphere from the ground up, while NRLMSISE-00 is a thermospheric model and is therefore only suitable for altitudes above 80-90km.
So to conclude, NRLMSISE takes into account the actual observations, while the Nasa standard atmosphere model is static, making NRLMSISE more accurate. For space applications like orbit determination, NRLMSISE's higher accuracy makes it much more suitable.
If you want an even better model however, I suggest you take a look at the Jacchia-Bowman 2008 model. It performs similar to NRLMSISE during normal solar activity levels but is more accurate during geomagnetic storms and during low solar activity periods. See this Phd thesis
for a comparison between atmospheric models (pg. 141).