Due to Precession, Nutation, and smaller movements caused by the earth spinning, the pole of the Earth changes position in the sky. Because of that, any coordinates in terms of Right Ascension and Declination will also move, since the Equatorial Coordinate system is referenced to the pole.
Epoch J2000 gives the coordinates of a point in the sky referenced to the position of the pole on the first day of year 2000 CE. JNOW is the coordinates of the same point, referenced to the pole at today's date and time.
ASIAIR uses JNOW, so you will need to use the current date for Stellarium's coordinates.
Chen