If you take 52 Cygni (in the Cygnus loop) as a reference star, since it is in the region you were looking at, its J2000 coordinates (according to SkySafari) is:
RA = 20h 46m 30s, DEC = +30º 47' 39.9"
Again, according to SkySafari (with nutation and precession), JNOW (right at this moment :-) is
RA = 20h 45m 39s, DEC = +30º 43' 11.6"
The difference from J2000.0 and JNOW is about 1m (15 arc minutes) different in RA, and about 4.5 arc minutes different in DEC.
Your numbers show pretty much the same difference between Astrometry.net and ASIAIR. (Another source of error could be how atmospheric refraction is estimated -- not knowing your latitude and observing time, I can't venture to guess if that is an additional source of error).
From the RA and DEC of stars in its database, we know that ASIAIR must use JNOW, since that is where the objects are when you are trying to point to them tonight, and not in January 2000 AD. Otherwise when we execute a GOTO in ASIAIR, we won't land on the correct spot. I think I have also seen ZWO folks confirm that the ASIAIR uses JNOW.
Astrometry.net uses J2000.0 according to this:
http://www.astro.umontreal.ca/atelieretudiant/Documents/JG_astrometrynet_manual.pdf
The difference between J2000.0 and JNOW appears to be consistent with the numbers you have obtained, so there is likely no error in plate solving by ASIAIR (at least not to the magnitude that you saw). It is probably just simply the case that ASIAIR is solving for the position your telescope is pointed to tonight, while Astrometry.net is solving for the position of stars as if it were January 2000 AD.
For what its worth, I had used the Hipparcos catalog (HYG 3.0) in my mount simulator (http://www.w7ay.net/site/Applications/MountSim/), which gives star coordinates referenced to J2000.0, and I had used nutation and precession (precession has quite a bit less delta than nutation, by the way) to convert J2000.0 to JNOW, so that the simulator displays what we see in the sky right now. When I tested ASIAIR's plate solve to estimate the center of my simulator's display, I usually came up with no more than a few arc seconds of error.
Chen