Could someone help me to understand the difference between the AP spectrum analyzer and the laptop-based spectrum analyzer and what's the best practice to use both of them?
Spectrum analysis (SA) requires special hardware. Not all wireless NICs can do SA. Some vendors have implemented SA functionality in their APs. When the AP is in SA mode it cannot act as an AP for clients.
The AP SA is good for displaying the radio spectrum as the AP sees it, because it is in the exact same location. However, you only see the spectrum at that single spot. If you need to find an interference source for example, a hand-held SA or laptop based SA with a special receiver is more mobile.
You didn't ask, but I'd also like to add that compared to laboratory based SA's, PC based versions are tailored to Wi-Fi channels and display more useful information, and are much more intuitive.
Obviously the size, weight, and cost are better too :-).
In RF dense environments, it can still take a lot of practice to identify an offenders modulation type, even with a PC based version.
The simplified displays from a non-overlapped DSSS, OFDM, or Bluetooth device, is easy to identify. But add in a multitude of devices, and it can be hard to identify/find offending devices. Some of the PC based tools, have virtual "device masks" which can help you identify a device type, but again these can be useless in a really busy environment.
Adding a good directional antenna to the mix really helps locate interference sources - if that is your goal. I prefer the flag shaped one from the Fluke/Netscout "AirCheck" WiFi Tester, more than the "Device Finder" model from MetaGeek. Although the MetaGeek version is a good little 7dBi antenna, it doesn't have a high enough Front-to-Back ratio to be labeled exceptional.