Both of them depend solely on the information reported to them during the life of the vehicle.
For example, my father had a 1993 GMC Sierra, picked it up new, put FARM USE tags on it, and drove it that way until 2005. Then Personal Use tags went on, when he decided to retire it from farm duty. In that time, it had gotten into 3 accidents, two due to farm equipment (minor) and one from a 18 wheeler not stopping quite fast enough (major.)
The back 2' of the frame was bent, the tailgate was ruined, and the back of the bed was caved in slightly, plus the frame was bent/cracked where the bed/cab meet slightly. Dad used a torch and cut the rear 2' of frame off, welded a new one in, replaced the bed, and no one was the wiser. None of it was reported, and none of it is on the CARFAX report. Infact, there is a 12 year gap where there is no information, due to it being a FARM USE truck. I sold it last year.
There are other numerous examples I could use, but this works best. CARFAX, AutoCheck, etc. are only as reliable as the data they are given in the first place. Sometimes everything checks out perfectly, sometimes not.