What advantages do CRTs still have over more modern display technologies?
CRT displays have several advantages over more modern technologies:  


Minimal response time, as incoming pixel data is generally beamed to the screen immediately as received, with virtually no buffering or complex processing like scaling. The phosphors also light up pretty much instantly when desired, unlike liquid crystal pixels.
Excellent black level (and contrast), especially when compared with older LCD technology.  
Resolution flexibility. As CRTs don't really have a fixed raster of pixels, they can scan the display at a wide range of vertical and horizontal resolutions without having to resort to scaling.
Arguably color rendition.  


Of course, the advantages didn't sufficiently offset the many disadvantages that CRTs have:  

Higher production cost, larger environmental impact (difficult to recycle, hazardous substances, high power consumption), susceptibility to screen burn-in, sensitivity to magnetic fields, significant warm-up time, tendency to flicker, imposing mass and volume and limited maximum screen size come to mind.