This^^^^
If I can expand on that excellent explanation...
If you're unaware of it, it's very disconcerting, and as TT described, completely counter-intuitive. It's also typically a function of the chartplotter, since the chartplotter is telling the AP how to steer (to a waypoint), that's really all the AP does. It uses compass heading to tell what direction to steer, rudder feedback so it knows how much it's steering compared to how far it's off course based on the compass heading (if it's steering an "AUTO" or "COMPASS" course), or cross track error (XTE) if it's steering a route to a waypoint. The AP doesn't know it's steering to a particular point, even though it repeats information about that waypoint- e.g. distance, bearing, course, etc. All that data is being calculated by the chartplotter and is transmitted to the AP- mostly for display.
The AP is essentially following the instruction of the chartplotter when steering to a waypoint, the
chartplotter is doing the heavy lifting. The AP changes to the next waypoint on the route because the chartplotter has determined that the waypoint has been reached, and it's time to change course.
And
SO..... all this discussion about which AP works better... well some of that depends on how it's used. If you use your AP to follow a route, then the chartplotter has more to do with how that works than the AP- to a degree. Some AP's have gyro function that makes them steer more naturally, some follow a compass heading more accurately, and some are easier to configure with rate of turn, rudder dampening, yaw, hunting, and response. But it all has to do with steering the boat.
My Raymarine AP follows instructions from Coastal Explorer, no problem. It will alternatively take instruction from the Furuno chartplotter. CE is smoother, hands down. Would the Raymarine AP work better with a Ray chartplotter?? Maybe. Some manufacturers utilize proprietary data that is transmitted between instruments. It usually takes priority, so there's a chance that "in-family" gear will perform better together. Disclaimer, my system is all NMEA, not N2K. Another topic.
That said, there's no reason an AP won't work with any other chartplotter, and if you're using a PC with software, it will work fine. The software can easily customize the data that it transmits to the AP, in fact CE is configured with the brand of AP in order to optimize the interface. I'm a CE fanboy, admittedly. My Furuno sits second chair as backup. As a full-time cruiser with over 38,000 NM, the majority of those miles have been from CE routes steered by the AP.
Best advice? Whatever your system,
learn how to use it. If you install it yourself, learn how to properly configure it, or enlist help from someone who can. If you pay to have it installed, ask about how it will be configured. If no mention of a sea trial or at-sea configuration, you haven't chosen the best installer. Don't pay all that money for gear that won't perform to its full potential.
apologies for the long winded reply.