The FAA does have specific regulations that could apply to making videos during part 135 and part 121 flights. As JScarry explained in his answer the part 135 rules may have been used against one prominent Youtuber.
135.100 says (emphasis mine):
§135.100 Flight crewmember duties.
(a) No certificate holder shall require, nor may any flight crewmember
perform, any duties during a critical phase of flight except those
duties required for the safe operation of the aircraft. Duties such as
company required calls made for such nonsafety related purposes as
ordering galley supplies and confirming passenger connections,
announcements made to passengers promoting the air carrier or
pointing out sights of interest, and filling out company payroll and related records are not required for the safe operation of the
(b) No flight crewmember may engage in, nor may any pilot in command
permit, any activity during a critical phase of flight which could
distract any flight crewmember from the performance of his or her
duties or which could interfere in any way with the proper conduct of
those duties. Activities such as eating meals, engaging in
nonessential conversations within the cockpit and nonessential
communications between the cabin and cockpit crews, and reading
publications not related to the proper conduct of the flight are not
required for the safe operation of the aircraft.
(c) For the purposes of this section, critical phases of flight
includes all ground operations involving taxi, takeoff and landing,
and all other flight operations conducted below 10,000 feet, except
For part 121, 121.542 says essentially the same thing, with an additional section banning the use of electronic devices at duty stations for non-flight purposes.
I remember reading or hearing somewhere (maybe reddit) that the FAA went after Steveo1kinevo because he was making videos during part 135 flights that included a lot of commentary during takeoff and landing, i.e. "nonessential conversations" during critical phases of flight.
In any case, clearly recording videos during takeoff and landing in 135/121 flights is no problem; there are hundreds of them on Youtube and airlines even use them in promotional materials. It seems very likely that the FAA's real concern is distraction, i.e. not just recording but actively participating in the video.