Pixhawk runs the ardupilot software, but if you're deciding between the Pixhawk hardware vs the APM hardware, I would go with the supported pixhawk hardware as APM is basically deprecated. The software is frozen for that hardware. No new features in arduplane/copter.
If cost is an issue, then that can't be helped as the APM stuff is pretty cheap now, but still functional.
I agree with makattack, while APMs are cheap and simple they're also defunct which means no further development or support is available, including compatible updates from ArduPilot.
If you want to stay in ArduPilot ecosystem (and you definitely should) then the Pixhawk and its variants are the way to go. Much faster processing, more features, more memory, SD card support, and it will remain viable for the foreseeable future.
I'd go with the Pixhawk. The features and capabilities are worth the extra cost. There are some reasonably priced ones on the market now too. You could also pick up a cheep APM and then upgrade the FC later.
I would disagree that APM is defunct. It remains one of the more popular fully-autonomous capable autopilots in existence, and is flown by 10's of thousands of users. I fly both Pixhawk and APM. I find APM much simpler to set up for ArduCopter (306 params vs 598 for the Pixhawk), but doesn't have all the bells and whistles of the Pixhawk for fully autonomous flight.
And development continues on APM, both on the hardware and software sides, but outside of the ArduPilot community. The last 3DR APM hardware was v2.6 with no onboard compass. The latest from third-party continued development is v2.8 with onboard compass, and a simple jumper to remove (no more cutting solder traces aka v2.5) by the GPS port to use external. And being it's based on Arduino, it continues to be the primary development platform for builders using the Arduino tools, including using APM for I/O interfaced with boards running linux.
Set up with a uBlox M8N with external compass I don't think you'd be disappointed in APM's performance and relative ease of setup for a quadrotor. Drop AC3.2.1 in it and it's a rock-stable platform that never crashes the OS or reboots in-flight, never have to worry about firmware updates breaking something, and has all of the features most people will ever need to fly a quad on fully autonomous GPS waypoint flights.
The larger flash capacity and processing power of the Pixhawk has made possible extra enhancements like EKF/EKF2, more output channels to handle auxiliary functions, more param settings for autonomous flight (ex: can set RTL speed separately, where APM uses WPNAV_SPEED), and possible to handle more advanced payload functions. But the 'hawk can also be overwhelming to a new user of the ArduPilot flight stack, and it is not plug-n-play. So if you're not willing to study the documentation for a couple weeks to learn what everything does, you're probably better off to start with an APM and get your feet wet with ArduPilot with that.
Painless360 has an excellent 8-video tutorial series on the APM hardware, setting it up and getting it ready for first flight: https://youtu.be/30cCs4aHdB0
I tend to say that if you want rock-solid stability with no surprises, and 'just works' every time, go with APM. If you are enamored by having every feature imaginable, running a MUCH larger and more complicated codebase with a Nuttx RTOS, and don't mind troubleshooting issues when you drop the latest firmware in, go with Pixhawk.
One poster mentioned NAZA-M V2, which I also have a lot of experience with. The NAZA is more plug-n-play but it is not even in the same universe as APM for autonomous flight capability. NAZA was developed for DJI's marketing sphere. The ArduPilot stack was built from the ground up for fully autonomous flight, and is by far the best (and most complicated) there is.
Another economical possibility for autonomous flight, with very easy setup, is the LibrePilot CC3D Revo (which I also fly). But the LibrePilot stack is not of the same pro-grade level as ArduPilot (at this point in development).